What is the integral of cos^5(x)sin(2x) ?

The integral of cos^5(x)sin(2x) is -2/7 cos^7(x)+C

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integral of cos^5(x)sin(2x)

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

- \frac{2}{7} cos^{7} (x) + C

Solution steps

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

Rewrite using trig identities

= \int 2 cos^{6} (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^{6} (x) sin (x) d x

Apply u-substitution

: \int - u^{6} d u

= 2 \cdot \int - u^{6} d u

Take the constant out:

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

= 2 (- \int u^{6} d u)

Apply the Power Rule

: \frac{u ^{7}}{7}

= 2 (- \frac{u ^{7}}{7})

Substitute back

u = cos (x)

= 2 (- \frac{cos ^{7} ( x )}{7})

Simplify

2 (- \frac{cos ^{7} ( x )}{7}) : - \frac{2}{7} cos^{7} (x)

= - \frac{2}{7} cos^{7} (x)

Add a constant to the solution

= - \frac{2}{7} cos^{7} (x) + C

\frac{\partial}{\partial x} (x^{2} - 2 x y + 6 y^{2} + x - 5 y)

\int y^{5} d y

e^{13 t}

x \to 0 lim (\frac{e ^{x} + 1}{x})

3 x^{2} - 5 x

What is the integral of cos^5(x)sin(2x) ?
The integral of cos^5(x)sin(2x) is -2/7 cos^7(x)+C