CUET UG 2025 — Mathematics Calculus

The integral is: $\displaystyle \int \frac{e^x(1 + x)dx}{\cos^2(e^x x)}$∫cos2(exx)ex(1+x)dx​

Notice that $xe^x$xex appears in the denominator inside $\cos^2(xe^x)$cos2(xex).

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Let $u = xe^x$u=xex

Using the product rule:

$$\frac{du}{dx} = e^x + xe^x$$dxdu​=ex+xex

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

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

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The integral becomes:

$$\int \frac{du}{\cos^2(u)}$$∫cos2(u)du​

$$= \int \sec^2(u)du$$=∫sec2(u)du

$$= \tan(u) + c$$=tan(u)+c

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Substituting $u = xe^x$u=xex:

$$\tan(xe^x) + c$$tan(xex)+c

where $c$c is the constant of integration.