If f(x)= e^x( x+^2 x) dx and f(0)=0, then f(()/4) equals: Option 1: 0 Option 2: -1 Option 3: 1 Option 4: e^(/4)

Answer: 1. Solution: Write x + ^2x = ( x - 1) + ^2 x, which is of the form g(x) + g'(x) with g(x) = x - 1. So f(x) = e^x( x - 1) + C. From f(0) = -1 + C = 0, we get C = 1. Then f(/4) = e^(/4)(1-1) + 1 = 1.

Calculus

Integrals

Medium

core

If $f(x)=\int e^x(\tan x+\tan^2 x)\,dx$ and $f(0)=0$ , then $f\left(\dfrac{\pi}{4}\right)$ equals:

-1

$e^{\pi/4}$

✅ Correct Option: 3

Write $\tan x + \tan^2x = (\tan x - 1) + \sec^2 x$ , which is of the form $g(x) + g'(x)$ with $g(x) = \tan x - 1$ .

So $f(x) = e^x(\tan x - 1) + C$ . From $f(0) = -1 + C = 0$ , we get $C = 1$ .

Then $f(\pi/4) = e^{\pi/4}(1-1) + 1 = 1$ .

2024: 16 May Shift 1

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

2022: 25 May Shift 1

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:

2025: 29 May Shift 2

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

2026: 25 May Shift 1

Integral $I = \displaystyle\int \dfrac{e^{3\log x} - 3x^4}{e^{2\log x} + x^2}\, dx$ (where $x>0$ ) is equal to

2025: 21 May Shift 2

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:

2026: 18 May Shift 2

The value of $\int_{-1}^{1} \dfrac{x^6}{6^x+1}\,dx$ is

If f(x)=∫ex(tan⁡x+tan⁡2x) dxf(x)=\int e^x(\tan x+\tan^2 x)\,dxf(x)=∫ex(tanx+tan2x)dx and f(0)=0f(0)=0f(0)=0, then f(π4)f\left(\dfrac{\pi}{4}\right)f(4π​) equals:

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

If $f(x)=\int e^x(\tan x+\tan^2 x)\,dx$ and $f(0)=0$ , then $f\left(\dfrac{\pi}{4}\right)$ equals: