Question

Evaluate the value of the following integral $\int{\dfrac{{{\sin }^{6}}x}{{{\cos }^{8}}x}dx}$

Answer 1

In this problem, we have to integrate the given integral expression. We can first simplify the terms using trigonometric identity. We can replace the term for $\dfrac{{{\sin }^{6}}x}{{{\cos }^{6}}x}$, we can then replace the remaining term with its equivalent secant term. We can then use the substitution formula and assume a variable for tangent and integrate it, to get the evaluated value of the given integral.

Complete step by step answer:
Here we have to integrate the given integral,
$\Rightarrow \int{\dfrac{{{\sin }^{6}}x}{{{\cos }^{8}}x}dx}$
We can now write it as,
$\Rightarrow \int{{{\tan }^{6}}x{{\sec }^{2}}xdx}$…… (1)
Since, $\dfrac{{{\sin }^{6}}x}{{{\cos }^{6}}x}={{\tan }^{6}}x,\dfrac{1}{{{\cos }^{2}}x}={{\sec }^{2}}x$
We can now integrate the above step using the substitution method.
We know that in the substitution method, we can assume a variable for the term in the integral and substitute the equivalent variable in the integral.
We can now assume the variable t.
Let $t=\tan x$..……. (2)
Then , $dt={{\sec }^{2}}xdx$
We can now substitute the above values in the integral (1), we get
$\Rightarrow \int{{{t}^{6}}dt}$
We can now integrate the above step, we get
$\Rightarrow \dfrac{{{t}^{7}}}{7}+C$
Since, $\int{{{t}^{n}}dt=\dfrac{{{t}^{n+1}}}{n+1}+C}$.
We can now substitute the value of t in the above step, we get
$\Rightarrow \dfrac{{{\tan }^{7}}x}{7}+C$
Therefore, $\int{\dfrac{{{\sin }^{6}}x}{{{\cos }^{8}}x}dx}=\dfrac{{{\tan }^{7}}x}{7}+C$

Note: We should always remember that if we have higher degrees for any trigonometric sines and cosines, we can use the trigonometric identities and formulas to simplify them and continue the problem, here we have used the formula, $\dfrac{{{\sin }^{n}}x}{{{\cos }^{n}}x}={{\tan }^{n}}x,\dfrac{1}{{{\cos }^{n}}x}={{\sec }^{n}}x$to simplify the step. We should also know some basic integral formulas like $\int{{{t}^{n}}dt=\dfrac{{{t}^{n+1}}}{n+1}+C}$, to integrate the given data.