Calculus

Integrals

Medium

applied

$\displaystyle\int \sqrt[3]{x}(x^3+4x^2+7)\,dx$ is equal to-

$3\left[13x^{\frac{3}{13}}+\dfrac{5}{2}x^{\frac{3}{10}}+\dfrac{4}{7}x^{\frac{3}{4}}\right]+c$ ,where $C$ is an arbitrary constant.

$3\left[\dfrac{1}{13}x^{\frac{13}{3}}+\dfrac{2}{5}x^{\frac{10}{3}}+\dfrac{7}{4}x^{\frac{4}{3}}\right]+c$ ,where $C$ is an arbitrary constant.

$\dfrac{1}{3}\left[\dfrac{1}{13}x^{13/3}+\dfrac{5}{2}x^{10/3}+\dfrac{7}{4}x^{4/3}\right]+c$ ,where $C$ is an arbitrary constant.

$3\left[13x^{\frac{3}{13}}+\dfrac{2}{5}x^{\frac{3}{10}}+\dfrac{4}{7}x^{\frac{3}{4}}\right]+c$ ,where $C$ is an arbitrary constant.

✅ Correct Option: 2

Expand the integrand as $x^{10/3}+4x^{7/3}+7x^{1/3}$ .

Integrating term by term gives $\dfrac{3}{13}x^{13/3}+\dfrac{6}{5}x^{10/3}+\dfrac{21}{4}x^{4/3}+c$ .

Taking $3$ common gives $3\left[\dfrac{1}{13}x^{13/3}+\dfrac{2}{5}x^{10/3}+\dfrac{7}{4}x^{4/3}\right]+c$ .

2025: 27 May Shift 1

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

2025: 3 June Shift 2

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) $\alpha = -1$

(D) $\beta = -1$

Choose the correct answer from the options given below:

2024: 16 May Shift 1

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

2026: 12 May Shift 2

Match List-I with List-II

List-I	List-II
Indefinite integral	Solution (where 'c' is an arbitrary constant)
(A) $\displaystyle\int \sqrt{x^2-a^2} \, dx$	(I) $\dfrac{1}{a}\tan^{-1}\dfrac{x}{a} + c$
(B) $\displaystyle\int \dfrac{1}{x^2-a^2} \, dx$	(II) $\dfrac{x}{2}\sqrt{x^2+a^2} + \dfrac{a^2}{2}\log\left[x+\sqrt{x^2+a^2}\right] + c$
(C) $\displaystyle\int \dfrac{1}{x^2+a^2} \, dx$	(III) $\dfrac{x}{2}\sqrt{x^2-a^2} - \dfrac{a^2}{2}\log\left[x+\sqrt{x^2-a^2}\right] + c$
(D) $\displaystyle\int \sqrt{x^2+a^2} \, dx$	(IV) $\dfrac{1}{2a}\log\left\vert \dfrac{x-a}{x+a} \right\vert + C$

Choose the correct answer from the options given below:

2022: 25 May Shift 1

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

2026: 22 May Shift 2

If $\int \frac{(\sqrt{x})^5}{(\sqrt{x})^7+x^6} dx = a\log\left|\frac{x^k}{1+x^k}\right| + C$ ; Where C is an arbitrary constant, then the value of $a.k$ is:

∫x3(x3+4x2+7) dx\displaystyle\int \sqrt[3]{x}(x^3+4x^2+7)\,dx∫3x​(x3+4x2+7)dx is equal to-

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

$\displaystyle\int \sqrt[3]{x}(x^3+4x^2+7)\,dx$ is equal to-