Exercise 7.3 · Q24 (MCQ): $\int\frac{e^x(1+x)}{\cos^2(e^x x)}\,dx$

Basics you need

Solution

1. Where is the integrand defined?

   We need $\cos(xe^x) \ne 0$⁠, that is,

   $$xe^x \ne \frac{\pi}{2} + n\pi, \quad n \text{ an integer}.$$

   The answer holds on any interval of $x$ on which this never fails. For example, on $0 \le x \le 0.7$ the function $xe^x$ increases from $0$ to $0.7e^{0.7} \approx 1.41$⁠, which is below $\frac{\pi}{2} \approx 1.57$⁠.

   $\frac{1}{\cos^2(xe^x)}$ blows up where the cosine is zero, so no antiderivative can cross those points. The numerator $e^x(1 + x)$ is defined everywhere.

2. Differentiate the inner function $xe^x$

   By the product rule with $u = x$ and $v = e^x$⁠:

   $$\begin{aligned} \frac{d}{dx}(xe^x) &= 1\cdot e^x + x\cdot e^x \\ &= e^x(1 + x). \end{aligned}$$

   This is exactly the numerator.

   Whenever a function sits inside another (here $xe^x$ inside $\cos^2$⁠), check whether its derivative appears elsewhere in the integrand. It does, with no extra constant, so the substitution $t = xe^x$ will absorb the numerator completely. (The book writes $e^x x$ inside the cosine; it is the same as $xe^x$⁠.)
   Product rule (Class XI; used throughout Chapter 5); $\frac{d}{dx}e^x = e^x$ (list in Section 7.2, (xi)).

3. Substitute $t = xe^x$ and integrate

   Let $t = xe^x$⁠. Then $dt = e^x(1 + x)\,dx$⁠, and

   $$\begin{aligned} \int\frac{e^x(1 + x)}{\cos^2(xe^x)}\,dx &= \int\frac{dt}{\cos^2 t} \\ &= \int\sec^2 t\,dt = \tan t + C \\ &= \tan(xe^x) + C. \end{aligned}$$
   The numerator times $dx$ becomes $dt$⁠, and the denominator becomes $\cos^2 t$⁠. Then $\frac{1}{\cos^2 t} = \sec^2 t$⁠, a standard integrand. Finally we put $t = xe^x$ back.
   Change of variable, Section 7.3.1; standard formula (iv), Section 7.2. Lesson: integration by substitution

4. Check: differentiate the answer back

   Chain rule with outer function $\tan u$ and inner function $u = xe^x$⁠:

   $$\begin{aligned} \frac{d}{dx}\tan(xe^x) &= \sec^2(xe^x)\cdot e^x(1 + x) \\ &= \frac{e^x(1 + x)}{\cos^2(xe^x)}. \end{aligned}$$

   This is the integrand. ✓

   Chain rule (Theorem 4, Section 5.3.1); Property (I), Section 7.2.1. Basics: integration undoes differentiation

5. Examine every option

   Differentiate each option and compare with the integrand. As a numerical test use $x = 0.5$⁠, where $xe^x \approx 0.824361$ and the integrand is about $5.363708$⁠.

   • (A) $-\cot(ex^x) + C$⁠. Whatever the inner function $u$⁠, the derivative of $-\cot u$ is $\operatorname{cosec}^2 u\cdot u' = \frac{u'}{\sin^2 u}$⁠: a sine squared in the denominator, not a cosine. Even if the printed $ex^x$ is read as $xe^x$⁠, the derivative is $\frac{e^x(1 + x)}{\sin^2(xe^x)} \approx 4.588933$ at $x = 0.5$⁠; read literally as $e\cdot x^x$⁠, it is about $0.669041$⁠. Neither is $5.363708$⁠. Wrong.
   • (B) $\frac{d}{dx}\tan(xe^x) = \frac{e^x(1 + x)}{\cos^2(xe^x)}$ for every $x$ in the domain (Step 4). Correct.
   • (C) $\frac{d}{dx}\tan(e^x) = \sec^2(e^x)\cdot e^x$⁠. The inner function is $e^x$⁠, not $xe^x$⁠, and the factor $1 + x$ is missing. At $x = 0.5$ it is about $272.065502$⁠. Wrong.
   • (D) The derivative of $\cot(e^x)$ is $-\operatorname{cosec}^2(e^x)\cdot e^x$⁠, which is negative for every $x$⁠, while the integrand is positive at $x = 0.5$ (about $-1.658773$ against $5.363708$⁠). Wrong.
   A wrong option needs only one point where its derivative differs from the integrand. The correct option is confirmed by the general calculation of Step 4.

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