Indefinite Integrals
10.0 Integral of type $\int {({{\sin }^m}x}$${\cos ^n}x)dx$
10.0 Integral of type $\int {({{\sin }^m}x}$${\cos ^n}x)dx$
In this type of integral both $m$ and $n$ are natural numbers. To solve this, different substitution are there depends on whether $m$ and $n$ are odd or even.
- If one of them is odd, then substitute for the term of even power i.e., if $m=5$ and $n=4$, then we substitute $\cos x = t$ and solve th integral using the methods explained earlier.
- If both are odd, then substitute either of them.
- If $\frac{{m + n}}{2} - 1$ is a negative integer, then substitute $\cot x = t$ or $\tan x = t$ according to the conditions given in the integral.
Question 10. Evaluate $$\int {{{\sin }^5}x} {\cos ^4}xdx$$
Solution: $$I = \int {{{\sin }^5}x} {\cos ^4}xdx$$ Put $\cos x = t \Rightarrow - \sin xdx = dt$, we get $$\begin{equation} \begin{aligned} I = - \int {{{\left( {1 - {t^2}} \right)}^2}.{t^4}dt} \\ I = - \int {\left( {{t^4} - 2{t^2} + 1} \right){t^4}} dt \\ I = - \int {\left( {{t^8} - 2{t^6} + {t^4}} \right)} dt \\ I = - \frac{{{t^9}}}{9} + 2\frac{{{t^7}}}{7} - \frac{{{t^5}}}{5} + C \\ I = - \frac{{{{\cos }^9}x}}{9} + 2\frac{{{{\cos }^7}x}}{7} - \frac{{{{\cos }^5}x}}{5} + C \\\end{aligned} \end{equation} $$
