Question

integration of sin to the power 5x

Answer 1

∫sin​5​​xdx

1

 

Use Trigonometric Reduction Formulas

-\frac{\sin^{4}x\cos{x}}{5}+\frac{4}{5}\int \sin^{3}x \, dx−​5​​sin​4​​xcosx​​+​5​​4​​∫sin​3​​xdx


2

 

Use Pythagorean Identities: \sin^{2}x=1-\cos^{2}xsin​2​​x=1−cos​2​​x

-\frac{\sin^{4}x\cos{x}}{5}+\frac{4}{5}\int (1-\cos^{2}x)\sin{x} \, dx−​5​​sin​4​​xcosx​​+​5​​4​​∫(1−cos​2​​x)sinxdx


3

 

Use Integration by Substitution on \int (1-\cos^{2}x)\sin{x} \, dx∫(1−cos​2​​x)sinxdx

Let u=\cos{x}u=cosx, du=-\sin{x} dxdu=−sinxdx


4

 

Using uu and dudu above, rewrite \int (1-\cos^{2}x)\sin{x} \, dx∫(1−cos​2​​x)sinxdx

\int -(1-{u}^{2}) \, du∫−(1−u​2​​)du


5

 

Use Power Rule: \int {x}^{n} \, dx=\frac{{x}^{n+1}}{n+1}+C∫x​n​​dx=​n+1​​x​n+1​​​​+C

\frac{{u}^{3}}{3}-u​3​​u​3​​​​−u


6

 

Substitute u=\cos{x}u=cosx back into the original integral

\frac{\cos^{3}x}{3}-\cos{x}​3​​cos​3​​x​​−cosx


7

 

Rewrite the integral with the completed substitution

-\frac{\sin^{4}x\cos{x}}{5}+\frac{4}{5}(\frac{\cos^{3}x}{3}-\cos{x})−​5​​sin​4​​xcosx​​+​5​​4​​(​3​​cos​3​​x​​−cosx)


8

 

Add constant

-\frac{\sin^{4}x\cos{x}}{5}+\frac{4}{5}(\frac{\cos^{3}x}{3}-\cos{x})+C−​5​​sin​4​​xcosx​​+​5​​4​​(​3​​cos​3​​x​​−cosx)+C


Done