Question

How do you integrate $\int {\dfrac{{cos(5x)}}{{{e^{\sin (5x)}}}}dx}$ using substitution?

Answer 1

First of all find the differentiation of the term given in the problem, $\sin (5x)$ with respect to $x$ and then use these values to substitute in the given integral and then find the integral after the substitution. Finally, re-substitute the value which was substituted in the obtained solution.

Complete step by step solution:
Consider the given integral as:
$I = \int {\dfrac{{\cos (5x)}}{{{e^{\sin (5x)}}}}dx}$
The goal of the problem is to find the integral using the substitution method.
Therefore, let us consider $t = \sin (5x)$
Differentiate both sides with respect to x.
$\dfrac{{dt}}{{dx}} = \dfrac{{d(\sin (5x))}}{{dx}}$
$\dfrac{{dt}}{{dx}} = 5\cos (5x)$
$\Rightarrow \cos (5x)dx = \dfrac{1}{5}dt$
Now, substitute the above obtained result in the integral I, so we have:
$I = \int {\dfrac{{dt}}{{{e^t}}}}$
As we know that $\int {{e^x}dx = {e^x}} + c$ we have
$I = \dfrac{{ - 1}}{{{e^t}}} + c$ , where c is the constant of integration.
Now, substitute the value of $t$ into the equation:
$I = \dfrac{{ - 1}}{{{e^{\sin (5x)}}}} + c$ , where c is the constant of integration.
Hence, this is the required result.
So, the correct answer is “ $I = \dfrac{{ - 1}}{{{e^{\sin (5x)}}}} + c$ ”.

Note : The integration by substitution is also said as “The reverse chain rule”.
This is the method to integrate in some special cases. Let $f(g(x))$ be the integrand and we have to find the integral of the function $\left[ {f(g(x))g'(x)} \right]$ , then we use this method of integration.
So, the integral is given as:
$\Rightarrow \int {\left[ {f(g(x))g'(x)} \right]dx}$
Now, assume that $g(x) = t$ and differentiate both sides with respect to $x$ .
$g'(x)dx = dt$
Now, make the substitution $g(x) = t$ and $g'(x)dx = dt$ in the integral.
$\Rightarrow \int {f(t)dt}$
Now, we can easily find the integral and re-substitute the value of $t$ in the resultant integral.