(e^x(1 + x)dx)/(^2(e^x x)) is equal to Option 1: (e^x) + c, where c is constant of integration. Option 2: -(ex^x) + c, where c is constant of integration. Option 3: (e^x) + c, where c is constant of integration. Option 4: (xe^x) + c, where c is constant of integration.

Answer: (xe^x) + c, where c is constant of integration. Solution: The integral is: (e^x(1 + x)dx)/(^2(e^x x)) Notice that xe^x appears in the denominator inside ^2(xe^x). Let u = xe^x Using the product rule: du/dx = e^x + xe^x du/dx = e^x(1 + x) du = e^x(1 + x)dx The integral becomes: du/(^2(u)) = ^2(u)du = (u) + c Substituting u = xe^x: (xe^x) + c where c is the constant of integration.

Calculus

Integrals

Hard

core

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

$\tan(e^x) + c$ , where $c$ is constant of integration.

$-\cot(ex^x) + c$ , where $c$ is constant of integration.

$\cot(e^x) + c$ , where $c$ is constant of integration.

$\tan(xe^x) + c$ , where $c$ is constant of integration.

✅ Correct Option: 4

The integral is: $\int \frac{e^x(1 + x)dx}{\cos^2(e^x x)}$

Notice that $xe^x$ appears in the denominator inside $\cos^2(xe^x)$ .

Let $u = xe^x$

Using the product rule:

$\frac{du}{dx} = e^x + xe^x$

$\frac{du}{dx} = e^x(1 + x)$

$du = e^x(1 + x)dx$

The integral becomes:

$\int \frac{du}{\cos^2(u)}$

$= \int \sec^2(u)du$

$= \tan(u) + c$

Substituting $u = xe^x$ :

$\tan(xe^x) + c$

where $c$ is the constant of integration.

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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:

∫ex(1+x)dxcos⁡2(exx)\int \frac{e^x(1 + x)dx}{\cos^2(e^x x)}∫cos2(exx)ex(1+x)dx​ is equal to

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

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