Question

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

Answer 1

Hint: 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.