What is the integral of \sqrt[4]{cos(x)}sin(x) ?

The integral of \sqrt[4]{cos(x)}sin(x) is -cos(x)\sqrt[4]{cos(x)}+1/5 cos^{5/4}(x)+C

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integral of \sqrt[4]{cos(x)}sin(x)

\int 4 cos (x) sin (x) d x

- cos (x) 4 cos (x) + \frac{1}{5} cos^{\frac{5}{4}} (x) + C

Solution steps

\int 4 cos (x) sin (x) d x

Apply Integration By Parts

: - cos (x) 4 cos (x) - \int \frac{1}{4} sin (x) cos^{\frac{1}{4}} (x) d x

= - cos (x) 4 cos (x) - \int \frac{1}{4} sin (x) cos^{\frac{1}{4}} (x) d x

\int \frac{1}{4} sin (x) cos^{\frac{1}{4}} (x) d x = - \frac{1}{5} cos^{\frac{5}{4}} (x)

= - cos (x) 4 cos (x) - (- \frac{1}{5} cos^{\frac{5}{4}} (x))

Simplify

= - cos (x) 4 cos (x) + \frac{1}{5} cos^{\frac{5}{4}} (x)

Add a constant to the solution

= - cos (x) 4 cos (x) + \frac{1}{5} cos^{\frac{5}{4}} (x) + C

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What is the integral of \sqrt[4]{cos(x)}sin(x) ?
The integral of \sqrt[4]{cos(x)}sin(x) is -cos(x)\sqrt[4]{cos(x)}+1/5 cos^{5/4}(x)+C