Q.7) $$\int \frac{e^x(1+x)}{\cos ^2\left(x e^x\right)} d x$$ is equal to

(A) $$\quad \tan \left(x e^x\right)+c$$

(B) $$\quad \cot \left(x e^x\right)+c$$

(C ) $$\quad \cot \left(\mathrm{e}^{\mathrm{x}}\right)+\mathrm{c}$$

(D) $$\quad \tan \left|\mathrm{e}^{\mathrm{x}}(1+\mathrm{x})\right|+\mathrm{c}$$

Solution:

Step 1: Use substitution method
Let $$t=x e^x$$.
Differentiating with respect to $$x$$ gives us:
$$
\frac{d t}{d x}=\frac{d}{d x}\left(x e^x\right)=1 \cdot e^x+x \cdot e^x=e^x(1+x)
$$

So, $$d t=e^x(1+x) d x$$.

Step 2: Rewrite the integral in terms of $$\boldsymbol{t}$$
Substitute $$t$$ and $$d t$$ into the original integral:
$$
\int \frac{e^x(1+x)}{\cos ^2\left(x e^x\right)} d x=\int \frac{d t}{\cos ^2(t)}=\int \sec ^2(t) d t
$$

Step 3: Integrate with respect to $$\boldsymbol{t}$$
The integral of $$\sec ^2(t)$$ is $$\tan (t)$$.
$$
\int \sec ^2(t) d t=\tan (t)+C
$$

Step 4: Substitute back for $$\boldsymbol{x}$$
Replace $$t$$ with $$x e^x$$ to get the final answer in terms of $$x$$ :
$$
\tan \left(x e^x\right)+C
$$
Answer: Correct option is ( a )