Q.Integrate the following function: $\int \frac{e^x (1 + x)}{\cos^2 (e^x x)} dx$ equals (A) $-\cot (e^x x) + C$ (B) $\tan (xe^x) + C$ (C) $\tan (e^x) + C$ (D) $\cot (e^x) + C$

The key idea is U Substitution: notice that the derivative of $e^x x$ appears in the numerator.

Let $u = e^x x$. Then $du = (e^x \cdot 1 + e^x \cdot x) dx = e^x (1 + x) dx$, which matches the numerator exactly.

The integral becomes:

$\int \frac{du}{\cos^2 u} = \int \sec^2 u \, du = \tan u + C.$ …

The key to this problem is seeing the structure. You have $e^x (1 + x)$ in the numerator and $\cos^2(e^x x)$ in the denominator. That denominator suggests something like $\sec^2(\text{something})$, which integrates to $\tan(\text{something})$. The question is: what is that "something"?

Notice that $e^x x$ is a product. Its derivative, using the product rule, is $e^x \cdot 1 + e^x \cdot x = e^x (1 + x)$. That is exactly the numerator. So the integrand is of the form $f'(x) \cdot \sec^2(f(x))$, where $f(x) = e^x x$.

This is a classic setup for the substitution $u = f(x)$.

1. Identify the substitution. Let $t = e^x x$. Then differentiate:

So $dt = e^x (1 + x) \, dx$, which is precisely the numerator times $dx$.

2. Rewrite the integral. The original integral is

Since $\frac{1}{\cos^2 t} = \sec^2 t$, we have

3. Integrate. The antiderivative of $\sec^2 t$ is $\tan t + C$. So

4. Substitute back. Replace $t$ with $e^x x$: …

Method: Spot a product-rule derivative in the argument, then substitute

For integrands like $\dfrac{e^x(1+x)}{\cos^2(e^x x)}$, recognise that the messy factor outside is the derivative of the argument inside, so a single substitution collapses everything.

Steps

Step 1: Identify the inner argument and differentiate it.

Let the argument be $g(x)=x\,e^x$. By the product rule,

— exactly the numerator.

Step 2: Substitute $u=g(x)$.

so the integral becomes $\displaystyle\int\frac{du}{\cos^2 u}=\int\sec^2 u\,du$. …

Common Mistakes

Mistake 1: Overlooking that $e^x(1+x)$ is the derivative of $x\,e^x$.

Why it's wrong: without spotting the product-rule derivative, the argument $x\,e^x$ inside $\cos^2$ looks impossible to handle. Correct approach: set $u=x\,e^x$, whose $du=e^x(1+x)\,dx$ is exactly the numerator.

Mistake 2: Misreading the argument as $e^x$ alone (options C/D). …