What is the integral of-2cos^3(x)(cos^2(x))''sin(x) ?

The integral of-2cos^3(x)(cos^2(x))''sin(x) is -2(-1/4 cos^4(x)(cos^2(x))''-1/3 cos^6(x))+C

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integral of-2cos^3(x)(cos^2(x))''sin(x)

\int - 2 cos^{3} (x) (cos^{2} (x))^{''} sin (x) d x

- 2 (- \frac{1}{4} cos^{4} (x) (cos^{2} (x))'' - \frac{1}{3} cos^{6} (x)) + C

Solution steps

\int - 2 cos^{3} (x) (cos^{2} (x))^{''} sin (x) d x

Take the constant out:

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

= - 2 \cdot \int cos^{3} (x) (cos^{2} (x))^{^{''}} sin (x) d x

Apply Integration By Parts

= - 2 (- \frac{1}{4} cos^{4} (x) (cos^{2} (x))^{^{''}} - \int - cos^{4} (x) sin (2 x) d x)

\int - cos^{4} (x) sin (2 x) d x = \frac{1}{3} cos^{6} (x)

= - 2 (- \frac{1}{4} cos^{4} (x) (cos^{2} (x))^{^{''}} - \frac{1}{3} cos^{6} (x))

Add a constant to the solution

= - 2 (- \frac{1}{4} cos^{4} (x) (cos^{2} (x))^{^{''}} - \frac{1}{3} cos^{6} (x)) + C

What is the integral of-2cos^3(x)(cos^2(x))''sin(x) ?
The integral of-2cos^3(x)(cos^2(x))''sin(x) is -2(-1/4 cos^4(x)(cos^2(x))''-1/3 cos^6(x))+C