Question

$\int \frac{\sin 2x \left(1 - \frac{3}{2}\cos x\right)}{e^{\sin^2 x + \cos^3 x}} \, dx =$

• A. $e^{\sin^{2}x+\cos^{3}x}+c$, where $c$ is a constant of integration.
• B. $-e^{-(\sin^{2}x+\cos^{3}x)}+c$, where $c$ is a constant of integration.
• C. $e^{-(\sin^{2}x+\cos^{3}x)^{2}}+c$, where $c$ is a constant of integration.
• D. $e^{\sin^{2}x+\cos~x}+c$, where $c$ is a constant of integration.

Hint:

Calculus Tip: In integrals featuring $e^{f(x)}$ in the denominator, taking $t = f(x)$ is almost always the correct first step. The numerator will usually simplify to exactly $dt$ or a scalar multiple of $dt$.

Answer 1

The Correct Option is B

Concept: Calculus - Integration by Substitution.

Step 1: Identify the complex term for substitution. The most complex part of the integral is the exponent of $e$ in the denominator. Let's substitute it: $t = \sin^{2}x + \cos^{3}x$.

Step 2: Differentiate the substitution to find $dt$. Differentiate $t$ with respect to $x$ using the chain rule: $dt = (2\sin x\cos x + 3\cos^{2}x(-\sin x))dx$.

Step 3: Simplify the derivative to match the numerator. Using the double angle formula $\sin 2x = 2\sin x\cos x$, substitute this into our differential equation: $dt = (\sin 2x - 3\sin x\cos^{2}x)dx$. Let's manipulate the given numerator to see if it matches. The numerator is $\sin 2x(1 - \frac{3}{2}\cos x)$. Expand this by replacing $\sin 2x$ with $2\sin x\cos x$ in the second term: $\sin 2x - (2\sin x\cos x)(\frac{3}{2}\cos x) = \sin 2x - 3\sin x\cos^{2}x$. This perfectly matches our $dt$.

Step 4: Rewrite and evaluate the integral in terms of $t$. Substitute $t$ and $dt$ back into the original integral: $\int\frac{1}{e^t}dt$. Rewrite it with a negative exponent: $\int e^{-t}dt$. The integration of $e^{-t}$ is simply $-e^{-t} + c$.

Step 5: Substitute $x$ back into the evaluated integral. Replace $t$ with our original substitution $(\sin^{2}x + \cos^{3}x)$ to get the final answer: $-e^{-(\sin^{2}x + \cos^{3}x)} + c$. $$ \therefore \text{The integral evaluates to } -e^{-(\sin^{2}x+\cos^{3}x)}+c. $$