What is the integral of-2picos^4(pix)sin^5(pix) ?

The integral of-2picos^4(pix)sin^5(pix) is -2(-1/5 cos^5(pix)+2/7 cos^7(pix)-1/9 cos^9(pix))+C

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integral of-2picos^4(pix)sin^5(pix)

\int - 2 π cos^{4} (π x) sin^{5} (π x) d x

- 2 (- \frac{1}{5} cos^{5} (π x) + \frac{2}{7} cos^{7} (π x) - \frac{1}{9} cos^{9} (π x)) + C

Solution steps

\int - 2 π cos^{4} (π x) sin^{5} (π x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - 2 π \cdot \int cos^{4} (π x) sin^{5} (π x) d x

Apply u-substitution

: \int cos^{4} (u) sin^{5} (u) \frac{1}{π} d u

= - 2 π \cdot \int cos^{4} (u) sin^{5} (u) \frac{1}{π} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - 2 π \frac{1}{π} \cdot \int cos^{4} (u) sin^{5} (u) d u

Rewrite using trig identities

= - 2 π \frac{1}{π} \cdot \int (1 - cos^{2} (u))^{2} sin (u) cos^{4} (u) d u

Apply u-substitution

: \int - v^{4} (1 - v^{2})^{2} d v

= - 2 π \frac{1}{π} \cdot \int - v^{4} (1 - v^{2})^{2} d v

Expand

- v^{4} (1 - v^{2})^{2} : - v^{4} + 2 v^{6} - v^{8}

= - 2 π \frac{1}{π} \cdot \int - v^{4} + 2 v^{6} - v^{8} d v

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - 2 π \frac{1}{π} (- \int v^{4} d v + \int 2 v^{6} d v - \int v^{8} d v)

\int v^{4} d v = \frac{v ^{5}}{5}

\int 2 v^{6} d v = \frac{2 v ^{7}}{7}

\int v^{8} d v = \frac{v ^{9}}{9}

= - 2 π \frac{1}{π} (- \frac{v ^{5}}{5} + \frac{2 v ^{7}}{7} - \frac{v ^{9}}{9})

Substitute back

= - 2 π \frac{1}{π} (- \frac{cos ^{5} ( π x )}{5} + \frac{2 cos ^{7} ( π x )}{7} - \frac{cos ^{9} ( π x )}{9})

Simplify

- 2 π \frac{1}{π} (- \frac{cos ^{5} ( π x )}{5} + \frac{2 cos ^{7} ( π x )}{7} - \frac{cos ^{9} ( π x )}{9}) : - 2 (- \frac{1}{5} cos^{5} (π x) + \frac{2}{7} cos^{7} (π x) - \frac{1}{9} cos^{9} (π x))

= - 2 (- \frac{1}{5} cos^{5} (π x) + \frac{2}{7} cos^{7} (π x) - \frac{1}{9} cos^{9} (π x))

Add a constant to the solution

= - 2 (- \frac{1}{5} cos^{5} (π x) + \frac{2}{7} cos^{7} (π x) - \frac{1}{9} cos^{9} (π x)) + C

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What is the integral of-2picos^4(pix)sin^5(pix) ?
The integral of-2picos^4(pix)sin^5(pix) is -2(-1/5 cos^5(pix)+2/7 cos^7(pix)-1/9 cos^9(pix))+C