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Exercise

$\int\sin^{9}x\cos^{3}xdx$

Step-by-step Solution

1

Apply the formula: $\int\sin\left(\theta \right)^n\cos\left(\theta \right)^mdx$$=\frac{-\sin\left(\theta \right)^{\left(n-1\right)}\cos\left(\theta \right)^{\left(m+1\right)}}{n+m}+\frac{n-1}{n+m}\int\sin\left(\theta \right)^{\left(n-2\right)}\cos\left(\theta \right)^mdx$, where $m=3$ and $n=9$

$\frac{-\sin\left(x\right)^{8}\cos\left(x\right)^{4}}{9+3}+\frac{9-1}{9+3}\int\sin\left(x\right)^{7}\cos\left(x\right)^3dx$
2

Simplify the expression

$\frac{-\sin\left(x\right)^{8}\cos\left(x\right)^{4}}{12}+\frac{2}{3}\int\sin\left(x\right)^{7}\cos\left(x\right)^3dx$
3

The integral $\frac{2}{3}\int\sin\left(x\right)^{7}\cos\left(x\right)^3dx$ results in: $-\frac{1}{15}\sin\left(x\right)^{6}\cos\left(x\right)^{4}-\frac{1}{20}\sin\left(x\right)^{4}\cos\left(x\right)^{4}+\frac{-\sin\left(x\right)^{2}\cos\left(x\right)^{4}}{30}+\frac{-\cos\left(x\right)^{4}}{60}$

$-\frac{1}{15}\sin\left(x\right)^{6}\cos\left(x\right)^{4}-\frac{1}{20}\sin\left(x\right)^{4}\cos\left(x\right)^{4}+\frac{-\sin\left(x\right)^{2}\cos\left(x\right)^{4}}{30}+\frac{-\cos\left(x\right)^{4}}{60}$
4

Gather the results of all integrals

$\frac{-\sin\left(x\right)^{8}\cos\left(x\right)^{4}}{12}+\frac{-\cos\left(x\right)^{4}}{60}+\frac{-\sin\left(x\right)^{2}\cos\left(x\right)^{4}}{30}-\frac{1}{20}\sin\left(x\right)^{4}\cos\left(x\right)^{4}-\frac{1}{15}\sin\left(x\right)^{6}\cos\left(x\right)^{4}$
5

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$

$\frac{-\sin\left(x\right)^{8}\cos\left(x\right)^{4}}{12}+\frac{-\cos\left(x\right)^{4}}{60}+\frac{-\sin\left(x\right)^{2}\cos\left(x\right)^{4}}{30}-\frac{1}{20}\sin\left(x\right)^{4}\cos\left(x\right)^{4}-\frac{1}{15}\sin\left(x\right)^{6}\cos\left(x\right)^{4}+C_0$

Final answer to the exercise

$\frac{-\sin\left(x\right)^{8}\cos\left(x\right)^{4}}{12}+\frac{-\cos\left(x\right)^{4}}{60}+\frac{-\sin\left(x\right)^{2}\cos\left(x\right)^{4}}{30}-\frac{1}{20}\sin\left(x\right)^{4}\cos\left(x\right)^{4}-\frac{1}{15}\sin\left(x\right)^{6}\cos\left(x\right)^{4}+C_0$

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  • Integrate using basic integrals
  • Product of Binomials with Common Term
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