Final Answer
$\frac{1}{4}\tan\left(x\right)^{4}-\ln\left(\cos\left(x\right)\right)-\frac{1}{2}\sec\left(x\right)^2+C_0$
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Step-by-step Solution
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Intermediate steps
1
Simplify $\frac{\sin\left(x\right)^5}{\cos\left(x\right)^5}$ into $\tan\left(x\right)^5$ by applying trigonometric identities
$\int\tan\left(x\right)^5dx$
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2
Applying a reduction formula for the integral of the tangent function: $\displaystyle\int\tan(x)^{n}dx=\frac{1}{n-1}\tan(x)^{n-1}-\int\tan(x)^{n-2}dx$
$\frac{1}{5-1}\tan\left(x\right)^{4}-\int\tan\left(x\right)^{3}dx$
Intermediate steps
3
Simplify the expression inside the integral
$\frac{1}{4}\tan\left(x\right)^{4}-\int\tan\left(x\right)^{3}dx$
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Intermediate steps
4
The integral $-\int\tan\left(x\right)^{3}dx$ results in: $-\frac{1}{2}\sec\left(x\right)^2-\ln\left(\cos\left(x\right)\right)$
$-\frac{1}{2}\sec\left(x\right)^2-\ln\left(\cos\left(x\right)\right)$
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5
Gather the results of all integrals
$\frac{1}{4}\tan\left(x\right)^{4}-\ln\left(\cos\left(x\right)\right)-\frac{1}{2}\sec\left(x\right)^2$
6
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
$\frac{1}{4}\tan\left(x\right)^{4}-\ln\left(\cos\left(x\right)\right)-\frac{1}{2}\sec\left(x\right)^2+C_0$
Final Answer
$\frac{1}{4}\tan\left(x\right)^{4}-\ln\left(\cos\left(x\right)\right)-\frac{1}{2}\sec\left(x\right)^2+C_0$