CUET Mathematics Calculus > Integrals PYQs

$\int_0^a \frac{\sqrt{x}}{\sqrt{x} + \sqrt{a-x}} dx$ is equal to

$\int_0^8 (x^{\frac{2}{3}} + 1) dx$ is equal to

$\int e^{2x}(\sin x + \frac{1}{2}\cos x) dx$ is equal to

If $\int \sqrt\frac{1-x}{{1+x}} dx = a\sqrt{1-x^2} + \beta \sin^{-1}x + C$, Where C is an arbitrary constant, then which of the following are TRUE? (A) $\alpha = 1$ (B) $\alpha = -1$ (C) $\beta = 1$ (D) $\beta = -1$ Choose the correct answer from the options given below:

Match List-I with List-II | List-I | List-II | | --- | --- | | (A) $\int \dfrac{dx}{x^2 - 16}$ | (I) $\dfrac{1}{8} \log \left\vert \dfrac{4 + x}{4 - x} \right\vert + c$, Where C is an arbitrary constant, | | (B) $\int \dfrac{dx}{x^2 + 16}$ | (II) $\log \left\vert x + \sqrt{x^2 - 16} \right\vert + c$, Where C is an arbitrary constant, | | (C) $\int \dfrac{dx}{16 - x^2}$ | (III) $\dfrac{1}{8} \log \left\vert \dfrac{x - 4}{x + 4} \right\vert + c$, Where C is an arbitrary constant, | | (D) $\int \dfrac{dx}{\sqrt{x^2 - 16}}$ | (IV) $\dfrac{1}{4} \tan^{-1} \left( \dfrac{x}{4} \right) + c$, Where C is an arbitrary constant, | Choose the correct answer from the options given below:

The value of $\int_2^4 \frac{x}{x^2 + 1} dx$ is equal to

$\int \sqrt{1 + \frac{x^2}{9}} dx$ is equal to (Where C is an arbitrary constant)

$\int_{1}^{2} \frac{\sqrt{x}}{\sqrt{3 - x} + \sqrt{x}} dx$ is equal to

$\int \frac{dx}{\sqrt{5 - 4x - x^2}}$ is equal to

$\int \frac{\log_e x}{(1 + \log_e x)^2} dx$ is equal to

Match List-I with List-II | List-I | List-II | | --- | --- | | Integral | Value | | --- | --- | | (A) $\int_{-1}^{1} (\vert x\vert + 1) dx$ | (I) 0 | | (B) $\int_{-2}^{2} \vert x + 1\vert dx$ | (II) 2 | | (C) $\int_{-1}^{1} 3\vert x^2\vert dx$ | (III) 5 | | (D) $\int_{-1}^{1} x\vert x\vert dx$ | (IV) 3 | Choose the correct answer from the options given below:

The value of $\int \frac{(x^4 - x)^{1/4}}{x^5} dx$ is equal to (where C is an arbitrary constant)

The value of $\int_0^1 \log_e\left(\frac{1}{x} - 1\right)dx$ is:

$\int \frac{e^{2x} - e^{-2x}}{e^{2x} + e^{-2x}}dx$ is equal to

$\int_{-1}^1 \frac{x^3 + |x| + 1}{x^2 + 2|x| + 1}dx$ is equal to

$\int_0^1 \frac{dx}{\sqrt{1+x} - \sqrt{x}}$ is equal to

The value of $\int_{-\pi/2}^{\pi/2}(x^5 + x^3\cos x)dx$ is

The value of $\int_{-a}^a f(x)dx$ where $f(x) = \frac{7^x}{1+7^x}$ is:

$\int e^{(x \log 5)}e^x dx$, is: Where $C$ is the constant of integration.

$\int \frac{1}{x(x^5-1)} dx$ is equal to

$\int\limits_{\sqrt{log_e 2}}^{\sqrt{log_e 4}} xe^{x^2} dx$ is equal to

The value of $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} (\sin|x| + \cos|x|)dx$, is equal to:

$\int \frac{\sqrt{16+(\log x)^2}}{x} dx$ is equal to (where C is an arbitrary constant)

Value of $\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \log(\tan x)dx$ is

$\int_1^{\sqrt{3}} \frac{1}{1+x^2} dx$ is equal to:

$\int (x^4 + x^2 + 1)d(x^2)$ is equal to: (where c is an integration constant)

$\int \frac{e^{7\log_e x} - e^{6\log_e x}}{e^{4\log_e x} - e^{3\log_e x}} dx$ is equal to: (Here, c is an arbitrary constant)

$\int_2^5 |x - 3|dx$ equals

Match List-I with List-II | List-I | List-II | |---|---| | (A) $\int_0^1 \frac{x^2}{1 + x^3} dx$ | (I) 0 | | (B) $\int_0^\pi 3\sin x dx$ | (II) $2\log_e\left(\frac{3}{2}\right)$ | | (C) $\int_{-1}^1 \sin^5 x \cos^6 x dx$ | (III) 6 | | (D) $\int_2^3 \frac{4}{x^2 - 1} dx$ | (IV) $\frac{1}{3}\log_e 2$ | Choose the correct answer from the options given below:

$\int \left(\frac{\cos 2x - \cos 2\alpha}{\cos x - \cos \alpha}\right) dx =$ (Given that $c$ is an arbitrary constant)

$\int_0^{\pi/2} \frac{\sin^8 x}{\sin^8 x + \cos^8 x} dx$ is equal to

$\int \frac{1}{(x + 1)(x + 2)} dx$ is equal to

The value of $\int_{-1}^{1}|x|dx$ is

$\int (e^{x\log a} + e^{a\log x}) dx$ is equal to (where $a > 1$)

Match List-I with List-II | List-I | List-II | | --- | --- | | Integral | Solution: C is an arbitrary constant | | --- | --- | | (A) $\int \frac{dx}{x^2 + 25}$ | (I) $\frac{1}{10} \log \left\vert \frac{5 + x}{5 - x} \right\vert + C$ | | (B) $\int \frac{dx}{x^2 - 25}$ | (II) $\log \vert x + \sqrt{x^2 - 25}\vert + C$ | | (C) $\int \frac{dx}{25 - x^2}$ | (III) $\frac{1}{5} \tan^{-1} \left( \frac{x}{5} \right) + C$ | | (D) $\int \frac{dx}{\sqrt{x^2 - 25}}$ | (IV) $\frac{1}{10} \log \left\vert \frac{x - 5}{x + 5} \right\vert + C$ | Choose the correct answer from the options given below:

$\int \frac{dx}{(1+5\sin^2 x)}$ is equal to

If $\int_0^a 3x^2dx = 8$, then the value of $a$ is:

The value of $\int \frac{x^5}{\sqrt{1 + x^3}} dx$ is

The value of $\int_{-1}^{1} |x^3 - x| dx$ is

$\int \frac{e^{2x} - 1}{e^{2x} + 1} dx =$

$\int e^{-x}(\cot x + \cosec^2 x)dx =$

$\int \frac{dx}{e^x + e^{-x}}$ is equal to

The value of $\int_{-5}^{5} |x + 3| dx$ is

$\int_{1}^{2} \frac{1}{x(x+1)} dx, x > 0$ equals

$\int \frac{(x-1)e^x}{x^2} dx, x > 0$ equals (where C is an arbitrary constant)

$\int e^x \left(\frac{1-x}{1+x^2}\right)^2 dx$ is equal to

If $f(x)$ and $g(x)$ are continuous functions in [0, a] such that $f(x) = f(a - x)$ and $g(x) + g(a - x) = a$ then $\int_{0}^{a} f(x)g(x)dx =$

The value of $\int_{0}^{\pi/2} \frac{\tan^7 x}{\cot^7 x + \tan^7 x} dx$ is

The integral $\int \frac{2dx}{e^{2x}-1}$ is equal to:

$\int \frac{\sin 2x \, dx}{\sqrt{9 - \cos^4 x}}$ equals

$\int_0^2 (|x| + |x - 2|) dx =$

If $\int_0^1 \frac{e^x}{1 + x} dx = m$, then the value of $\int_0^1 \frac{e^x}{(1 + x)^2} dx$ is:

$\int \frac{dx}{x^3\sqrt{(1 + x^4)}} =$

Match List-I with List-II (where $c$ is an arbitrary constant) | List-I | List-II | | --- | --- | | (A) $\int \tan x \, dx$ | (I) $\log\vert \sec x + \tan x\vert + c$ | | (B) $\int \cot x \, dx$ | (II) $\log\vert \sec x\vert + c$ | | (C) $\int \sec x \, dx$ | (III) $\log\vert \sin x\vert + c$ | | (D) $\int \cosec x \, dx$ | (IV) $\log\vert \cosec x - \cot x\vert + c$ | Choose the correct answer from the options given below:

If $\int \frac{2x - 5}{(2x - 3)^3} e^{2x} dx = \frac{\lambda e^{2x}}{(2x - 3)^2} + C$, where $C$ is an arbitrary constant then the value of $\lambda$ is

$\int \frac{x}{(x-1)(x-2)} dx$ is equal to ( where $C$ is a constant of integration)

$\int_0^1 x e^x dx$ is equal to

$\int \frac{dx}{2\sin^2 x + 5\cos^2 x}$ is equal to

If $f(a-x) = f(x)$, then $\int_0^a xf(x)dx$ is equal to

$\int \left(\frac{1}{\log_e x} - \frac{1}{(\log_e x)^2}\right)dx$ is equal to

The Value of $\int_1^3 |2x - 1|dx$ equal to

The integral $\int e^x\left(\frac{x-1}{2x^2}\right)dx$ is equal to

Value of $\int \left(\frac{1}{logx} - \frac{1}{(logx)^2}\right)dx$ is

The value of $\int_1^3 \frac{x^2}{x^3+1}dx$

Value of $\int_2^3 \frac{\sqrt{x}}{\sqrt{x} + \sqrt{5-x}}dx$ is

$\int_0^{\pi/2} \sqrt{1 - \sin 2x}\,dx$ is equal to:

$\int \frac{\sin x - x\cos x}{x(x + \sin x)}dx =$ (where C is an arbitrary constant)

If $\int \frac{x^4}{x-2}dx = px + qx^2 + rx^3 + sx^4 + t\log |x - 2| + C$, where C is an arbitrary constant and p, q, r, s, t are real numbers, then the correct arrangement of p, q, r, s, t is:

The value of the definite integral $\int_0^1 e^x \frac{(1-x)^2}{(1+x^2)^2}dx$ is:

For $x > e$, $\int \frac{dx}{x - \sqrt{x}}$ is equal to

Match List-I with List-II | List-I | List-II | | --- | --- | | Definite integral | Value | | --- | --- | | (A) $\int_{1}^{e} \frac{\log x}{x} dx$ | (I) 4 | | (B) $\int_{-2}^{2} x^3(1 - x^2) dx$ | (II) $\frac{1}{2}$ | | (C) $\int_{1}^{2} x \, dx$ | (III) 0 | | (D) $\int_{-2}^{2} \vert x\vert dx$ | (IV) $\frac{3}{2}$ | Choose the correct answer from the options given below:

The value of $\int \left\{ \frac{1}{\log_e x} - \frac{1}{(\log_e x)^2} \right\} dx$ is

The value of $\int_0^{\pi/2} \log_e \left(\frac{5 + 2 \sin x}{5 + 2 \cos x}\right) dx$ is

Match List-I with List-II (Given that $c$ is an arbitrary constant) | List-I | List-II | | --- | --- | | (A) $\int \frac{dx}{\sqrt{a^2 - x^2}} =$ | (I) $\log_e \vert x + \sqrt{x^2 - a^2}\vert + c$ | | (B) $\int \sqrt{a^2 - x^2} dx =$ | (II) $\sin^{-1} \frac{x}{a} + c$ | | (C) $\int \sqrt{x^2 - a^2} dx =$ | (III) $\frac{x}{2} \sqrt{a^2 - x^2} + \frac{a^2}{2} \sin^{-1} \frac{x}{a} + c$ | | (D) $\int \frac{dx}{\sqrt{x^2 - a^2}} =$ | (IV) $\frac{x}{2} \sqrt{x^2 - a^2} - \frac{a^2}{2} \log_e \vert x + \sqrt{x^2 - a^2}\vert + c$ | Choose the correct answer from the options given below:

The value of $\int_0^1 x e^x dx$ is:

Value of $\int \frac{2}{(x-3)\sqrt{x+1}} dx$ is: (Here C is an arbitrary constant)

If $2f(x) + f\left(\frac{1}{x}\right) = x^2 + 1$, then $\int f(x) dx$ is: (Here C is an arbitrary constant)

Match List-I with List-II | List-I | List-II | |---|---| | (A) $\int_{-a}^a f(x) dx = 0$ | (I) 0 | | (B) $\int_0^{2a} f(x) dx = 2\int_0^a f(x) dx$ | (II) 1 | | (C) $\int_{-\pi}^{\pi} \cos x dx$ | (III) $f$ is an odd function | | (D) $\int_{-1}^1 x^{101} dx + 1$ | (IV) $f(2a-x) = f(x)$ | Choose the correct answer from the options given below:

$\int \tan^{-1}\sqrt{x} $ $dx$ equals to: (Here C is an arbitrary constant)

The value of the definite integral $I = \int_0^2 x\sqrt{2-x} dx$ is:

The value of $\int_0^1 [\log x - \log(1-x)] dx$ is

$\int \frac{(x-3)e^x}{(x-1)^3} dx$ is equal to

Match List-I with List-II | List-I | List-II | | --- | --- | | Functions | Integrals | | --- | --- | | (A) $\int \frac{dx}{x^2 - 4}, x \neq \pm 2$ | (I) $\log \vert x + \sqrt{4 + x^2}\vert + C$, where $C$ is an arbitrary constant | | (B) $\int \frac{1}{\sqrt{16 - x^2}} dx; \vert x\vert < 4$ | (II) $\sin^{-1} \left( \frac{x}{4} \right) + C$, where $C$ is an arbitrary constant | | (C) $\int \frac{1}{16 + x^2} dx$ | (III) $\frac{1}{4} \log \left\vert \frac{x-2}{x+2} \right\vert + C$, where $C$ is an arbitrary constant | | (D) $\int \frac{1}{\sqrt{4 + x^2}} dx$ | (IV) $\frac{1}{4} \tan^{-1} \left( \frac{x}{4} \right) + C$, where $C$ is an arbitrary constant | Choose the correct answer from the options given below:

$\int \frac{\cos 2x - \cos 2α}{\cos x - \cos α} dx$ is equal to

$\int_{3\pi/8}^{\pi/8} \frac{ \tan^{2025} x}{ \tan^{2025} x + \cot^{2025} x} dx$ is equal to

If the integral $I = \int \frac{x^2}{\sqrt{1+x}} dx = \frac{1}{\alpha} (1+x)^\alpha - \frac{8\alpha}{15} (1+x)^{\alpha-1} + 2(1+x)^{\alpha-2} + C$, $C$ is constant of integration, then the value of $\alpha$ is:

$\int \frac{f'(x)}{f(x) \log_e[f(x)]} dx$ is equal to

$\int \frac{(x^4 - x)^{1/4}}{x^5} dx$ is equal to

The value of $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} (\sin|x| + \cos|x|) dx$ is

Match List-I with List-II [.] denotes the greatest integer function. | List-I | List-II | |---|---| | (A) $\int_0^3 [x]dx$ | (I) $\frac{1}{2}$ | | (B) $\int_0^1 [2x]dx$ | (II) 1 | | (C) $\int_0^1 [3x]dx$ | (III) $\frac{3}{2}$ | | (D) $\int_0^1 [4x]dx$ | (IV) 3 | Choose the correct answer from the options given below:

For $x > 1$, $\int \frac{e^{7\log x} - e^{5\log x}}{e^{5\log x} - e^{4\log x}} dx$ equals.

The value of the definite integral $I = \int_{-1}^{1} \frac{1}{1 + \sqrt{e^x}} dx$ is:

If $I = \int \frac{x}{x - \sqrt{x^2 - 4}} dx = \alpha x^3 + \beta(x^2 - 4)^{\frac{3}{2}} + \gamma$, where $\gamma$ is constant of integration, then

Match List-I with List-II | List-I | List-II | | --- | --- | | (A) The value of $\int_{0}^{4} \vert x\vert \, dx$ is | (I) 3 | | (B) The value of $\int_{-2}^{2} \vert x\vert \, dx$ is | (II) -1 | | (C) The value of $\int_{0}^{3} [x] \, dx$ is | (III) 8 | | (D) The value of $\int_{-1}^{1} [x] \, dx$ is | (IV) 4 | Choose the correct answer from the options given below:

For $x \neq -1$, if $\int \frac{xe^x dx}{(1+x)^2} = \frac{ae^x}{(1+x)^b} + c$, where a, b are fixed numbers and c is the integration constant, then $a + b$ is equal to

For $x \in \left(0, \frac{\pi}{2}\right)$, $\int \frac{\sin x + \cos x}{\sqrt{\sin 2x}} dx$ is equal to

If $\int e^x\left(\frac{x-1}{(x+1)^3}\right)dx = \frac{Ae^x}{(x+1)^B} + C$, where C is constant of integration then which of the following are correct? (A) $A = -1$ (B) $A = 1$ (C) $B = 3$ (D) $B = 2$ Choose the correct answer from the options given below:

For $x \in \left(0, \frac{\pi}{2}\right)$, $\int \frac{1}{\sin^2 x + \sin 2x} dx$ is equal to

If $I = \int \frac{x^4 + x^2 + 1}{x^2 - x + 1} dx = \alpha x + \beta x^2 + \gamma x^3 + \delta$, $\delta$ is constant of integration, then $(\alpha + 2\beta + 3\gamma)$ equals

The value of the integral $I = \int_0^1 \frac{1}{\sqrt{x^2 + 2x + 3}} dx$ is:

For $x \in \mathbb{R} - \{-1,0,1\}$, $\int \frac{1}{x - x^5}dx$ is equal to

$\int_{-1}^{1}(x^7 + x^5 + x^3 + x + 1)dx$ is equal to

$\int \frac{e^x(1 + x)dx}{\cos^2(e^x x)}$ is equal to

$\int_{-\pi}^{\pi} \frac{e^{\sin x}}{e^{\sin x} + e^{-\sin x}}dx$ is equal to

If $I_n = \int_{0}^{\pi/4} \tan^n x dx$ then $I_{2024} + I_{2026}$ is equal to:

If $\int \frac{dx}{(x-1)^3/^4. (x+2)^5/^4} = a[1 - g(x)]^b + c$, where $c$ is a constant of integration, then which of the following are true? (A) $a = \frac{2}{3}$ (B) $\beta = \frac{3}{4}$ (C) $3\alpha + 4\beta = 5$ (D) $g(x) = \frac{3}{(x+2)}$ Choose the **correct** answer from the options given below:

The value of the definite integral $I = \int_{1}^{2} \frac{1}{x(1 + x^2)}dx$ is:

$\int_1^4 |x - 2|dx$ is equal to

The integral I = $\int \frac{e^{5\log_e x} - e^{4\log_e x}}{e^{3\log_e x} - e^{2\log_e x}} dx$ is equal to

Match List-I with List-II | List-I | List-II | |---|---| | Definite integral | Value | | (A) $\int_0^1 \frac{2x}{1 + x^2} dx$ | (I) 2 | | (B) $\int_{-1}^1 sin^3 x \cos^4 x dx$ | (II) $log_e\left(\frac{3}{2}\right)$ | | (C) $\int_0^{\pi} \sin x dx$ | (III) $log_e 2$ | | (D) $\int_2^3 \frac{2}{x^2 - 1} dx$ | (IV) 0 | Choose the correct answer from the options given below:

The integral I = $\int e^x\left(\frac{x - 1}{3x^2}\right) dx$ is equal to

$\int_{\pi/6}^{\pi/3} \frac{tan x}{tan x + cot x} dx$ is equal to

If $\int \frac{(1 + x \log x)}{xe^{-x}} dx = e^x f(x) + C$, where C is constant of integration, then $f(x)$ is

$\int \left(\frac{1}{log_e t} - \frac{1}{(log_e t)^2}\right) dt$ is equal to

$\int_0^2 x(2-x)^n dx$ is equal to

$\int_0^1 tan^{-1}\left(\frac{2x-1}{1+x-x^2}\right)dx$ is equal to

$\int_{-\frac{5}{2}}^{\frac{5}{2}} |x| dx$ is equal to

$\int \left(\frac{\cos x - \sin x}{1 + \sin 2x}\right) dx$ is equal to

If the integral $I = \int \left[log_e(log_e x)^2 + \frac{a}{log_e x}\right] dx = x log_e(log_e x)^2 + C$, where C is constant of integration. Then the value of $a$ is:

$\int \frac{x^3 - 1}{x^2} dx$ is equal to

The value of which of the following integrals is zero? (A) $\int_0^1 x dx$ (B) $\int_{-1}^1 x dx$ (C) $\int_{-1}^1 x^2 dx$ (D) $\int_0^1 \log\left(\frac{x}{1-x}\right) dx$ Choose the correct answer from the options given below:

$\int \frac{dx}{9x^2 - 16}$ is equal to

$\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{dx}{1 + \sqrt{tanx}}$ is equal to

$\int \sin x \sin 2x \sin 3x dx$ is equal to

The value of the integral $I = \int_0^1 \frac{1}{\sqrt{1+3\sqrt{x}}} dx$ is

$\int \dfrac{\pi}{x^{n+1}-x} d x=$

The value of $\int_{0}^{1} \frac{a-b x^{2}}{\left(a+b x^{2}\right)^{2}} d x$ is :

The value of the integral $\int_{\log _{e} 2}^{\log _{e} 3} \frac{e^{2 x}-1}{e^{2 x}+1} d x$ is :

$\int_{0}^{\frac{\pi}{2}} \frac{1-\cot x}{\operatorname{cosec} x+\cos x} d x=$

$\int \mathrm{e}^{\mathrm{x}}\left(\frac{2 \mathrm{x}+1}{2 \sqrt{\mathrm{x}}}\right) \mathrm{dx}$

The integral $\int e^x \left(\frac{x-1}{2x^2}\right) dx$ is equal to:

The value of $\int \left(\sqrt{x} + \frac{1}{\sqrt{x}}\right)^2 dx$ is:

$\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{dx}{1 + \sqrt{\tan x}}$ is equal to :

The value of integral $\int \sqrt{4x^2 + 9}\, dx$ is

$\int_1^2 \frac{x \, dx}{(x+1)(x+2)} =$

$\int e^x \left(\frac{1-x}{1+x^2}\right)^2 dx =$

$\int_0^{\pi/2} \sqrt{1 - \sin 2x} \, dx$ is equal to :

$\int e^x (\tan x + \log_e \sec x) \, dx =$

$\int \left(x + \frac{1}{x}\right)^2 dx$ equals :

$\int \frac{\sqrt{\tan x}}{\sin x \cos x} dx$ equals :

$\int e^x \sec x (1 + \tan x) dx$ equals :

The value of the integral $\int_{2}^{4} \frac{x}{x^2+1} dx$ is :

The value of $\int_{0}^{3} |2x - 6| dx$ is :

The integral $\int \frac{dx}{x^2(x^4+1)^{\frac{3}{4}}}$ equals __________.

The integral $\int_{0}^{1} x(1-x)^n dx$ is equal to :

$\int_{0}^{1} x^2 e^{x^3} \, dx$ is equal to :

If $\frac{d}{dx}(f(x)) = 5x^4 - \frac{4}{x^5}$ such that $f(1) = 0$. Then $f(2) - 2f\left(\frac{1}{2}\right)$ is equal to :

If $f(x) = \begin{cases} 1 - 2x, & x \leq 0 \\ 1 + 2x, & x > 0 \end{cases}$, then $\int_{-1}^{1} f(x) \, dx =$

$\int_{1}^{5} |x - 2| \, dx =$

$\int \frac{1}{\cos^2 x (1 + \tan x)^3} \, dx =$

$\int \frac{x^2 + 4}{(x^2 + 3)(x^2 + 5)} \, dx = u \tan^{-1}\left(\frac{x}{\sqrt{3}}\right) + v \tan^{-1}\left(\frac{x}{\sqrt{5}}\right) + c$, where c is arbitrary constant, then value of $\frac{1}{u^2} + \frac{1}{v^2}$ is equal to :

For $a, b \in R$ and $a < b$, $\int_{a}^{b} \frac{f(x)}{f(x) + f(a + b - x)} \, dx =$

If $\int(\sqrt{x+1} + \sqrt{x-1})^2 dx = \alpha x^2 + \beta x\sqrt{x^2-1} + \gamma \log|x+\sqrt{x^2-1}| + C$, then value of $\alpha + \beta - 2\gamma$ is :

If $g(x) = \int \frac{dx}{x^{1/2} + x^{1/6}}$, then $g(1) - g(0)$ is :

The value of $\int_0^1 e^x (x + 1) \, dx$ is equal to

If $f'(x) = 4x^5 - 6x$ and $f(0) = 3$, then $f(3)$ is equal to

The value of $\int_{-4}^{4} \log_e \left( \frac{1-x}{1+x} \right) dx$ is equal to

$\int \frac{x}{(x^2+3)(x^2+4)} dx =$

$\int e^x \left(\frac{1}{x} - \frac{2}{x^3}\right) dx =$

Match List I with List - II | List - I | List - II | |---|---| | A. An even function | I. $x^2 + \cos x$ | | B. For an even function, $\int_{-a}^{a} f(x)dx =$ | II. 0 | | C. If $f(2a-x) = -f(x)$, then $\int_{0}^{2a} f(x)dx =$ | III. $2\int_{0}^{a} f(x)dx$ | | D. An odd function | IV. $x^3 + \sin x$ | Choose the correct answer from the options given below:

$\int_{0}^{\pi} \frac{e^{\cos x}}{e^{\cos x} + e^{-\cos x}} dx =$

If $\int (x + \sqrt{x^2 - 1})^2 \, dx = \alpha \cdot x + \beta x^3 + \gamma (x^2 - 1)^{\frac{3}{2}} + C$, where $C$ is arbitrary constant, then the value of $3(\alpha + \beta + \gamma)$ is

$\int_0^1 \frac{dx}{x^2 + x + 1}$

Match List I with List II | List - I | List - II | |---|---| | A. $\int_{-\pi/2}^{\pi/2} \sin^7 x \, dx$ | I. $\frac{\pi}{2}$ | | B. $\int_{-\pi/2}^{\pi/2} \sin^2 x \, dx$ | II. $\frac{\pi}{4}$ | | C. $\int_0^{\pi/2} \frac{1}{1 + \tan x} \, dx$ | III. 0 | | D. $\int_0^{\pi} \lvert \cos x \rvert \, dx$ | IV. 2 |

$\int e^x \left(\frac{1-x}{1+x^2}\right)^2 dx =$

$\int \frac{\sin(\tan^{-1} x)}{1 + x^2} dx =$

$\int \frac{x^2 + 1}{x^4 + 1} dx =$

$\int_0^{4\pi} \frac{x}{1 + |\cos x|} dx =$

If $\int_0^{\pi/2} \sqrt{\tan x} \, dx = \frac{\lambda}{\sqrt{2}}$, then the value of $\lambda$ is

$\int \frac{xe^x}{(x+1)^2} dx =$

$\int_{\frac{1}{3}}^{1} \frac{(x - x^3)^{\frac{1}{3}}}{x^4} dx =$

$\int \sqrt{1 - 49x^2} \, dx$ is equal to

$\int x\sqrt{x + 2} \, dx$ is equal to :

The value of $\int_{-3}^{3} \log_e\left(\frac{4+x}{4-x}\right) dx$ is

$\int \frac{x^2+1}{(x+1)^2} e^x \, dx$ is equal to

$\int \frac{x}{(x-1)(x-2)} dx =$ (where c is an arbitrary constant)

The value of $\int_{\pi/2}^{\pi} \frac{1}{1 + \cot x} dx$ is equal to:

Match List I with List II | List I | List II | |---|---| | A. $\int \frac{dx}{x^2 - a^2}$ is equal to | I. $\frac{x}{2}\sqrt{x^2 + a^2} + \frac{a^2}{2}\log\left\lvert x + \sqrt{x^2 + a^2}\right\rvert + C$ | | B. $\int \frac{dx}{\sqrt{x^2 + a^2}}$ is equal to | II. $\frac{1}{2a}\log\left\lvert \frac{x-a}{x+a}\right\rvert + C$ | | C. $\int \sqrt{a^2 - x^2} \, dx$ is equal to | III. $\frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2}\sin^{-1}\frac{x}{a} + C$ | | D. $\int \sqrt{a^2 + x^2} \, dx$ is equal to | IV. $\log\left\lvert x + \sqrt{x^2 + a^2}\right\rvert + C$ | Choose the correct answer from the options given below:

$\int \frac{dx}{(e^x - 1)} =$

If $\int (1 + e^{-x} + e^{-2x} + ...)dx = \log\phi(x) + C$, then $\phi(x)$ is equal to :

Let $\int \frac{dx}{\left(\sqrt{x} - \sqrt{x-1}\right)^2} = \alpha u(x) + \beta v(x) + C$, where $u(x) = x^2 - x + \left(x - \frac{1}{2}\right)\sqrt{x^2 - x}$ and $v(x) = \log\left|x - \frac{1}{2} + \sqrt{x^2 - x}\right|$. The value of $\alpha + \beta$ is :

$\int \frac{x^2 - 4}{(x+2)(x-1)(x-3)} dx =$

$\int_{-\pi/2}^{\pi/2} \sin^7 x \, dx =$

$\int e^x \left(\frac{1}{x} - \frac{1}{x^2}\right) dx =$

$\int \frac{dx}{\sin^2 x \cos^2 x} =$

$\int \sqrt{x^2 - 4x + 5} \, dx =$

Match List I with List II | List - I | List - II | |---|---| | A. $\int \frac{dx}{x^2 - a^2} =$ | I. $\log_e\left\lvert x + \sqrt{x^2 + a^2}\right\rvert + C$ | | B. $\int \frac{dx}{a^2 - x^2} =$ | II. $\frac{1}{2a}\log_e\left\lvert\frac{x-a}{x+a}\right\rvert + C$ | | C. $\int \frac{dx}{\sqrt{x^2 - a^2}} =$ | III. $\frac{1}{2a}\log_e\left\lvert\frac{a+x}{a-x}\right\rvert + C$ | | D. $\int \frac{dx}{\sqrt{x^2 + a^2}} =$ | IV. $\log_e\left\lvert x + \sqrt{x^2 - a^2}\right\rvert + C$ | Choose the correct answer from the options given below:

$\int_{0}^{1.5} [x] dx$, where $[x]$ denotes the greatest integer function $\leq x$, is equal to :

$\int \left(\frac{1+x+x^2}{1+x^2}\right) e^{\tan^{-1} x} dx =$

Match List I with List II | LIST I | LIST II | |---|---| | A. $\int \frac{\sin x}{1 + \cos x} \, dx$ | I. $e^{\tan^{-1} x} + C$ | | B. $\int \frac{1}{1 - \tan x} \, dx$ | II. $\log(\log x + 1) + C$ | | C. $\int \frac{e^{\tan^{-1} x}}{1 + x^2} \, dx$ | III. $-\log\lvert 1+\cos x \rvert + C$ | | D. $\int \frac{1}{x + x \log x} \, dx$ | IV. $\frac{x}{2} - \frac{1}{2}\log\lvert \cos x - \sin x \rvert + C$ | Choose the correct answer from the options given below: