What is the integral of sin(5x)cos^{4/3}(5x) ?

The integral of sin(5x)cos^{4/3}(5x) is -3/35 cos^{7/3}(5x)+C

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integral of sin(5x)cos^{4/3}(5x)

\int sin (5 x) cos^{\frac{4}{3}} (5 x) d x

- \frac{3}{35} cos^{\frac{7}{3}} (5 x) + C

Solution steps

\int sin (5 x) cos^{\frac{4}{3}} (5 x) d x

Apply u-substitution

: \int sin (u) cos^{\frac{4}{3}} (u) \frac{1}{5} d u

= \int sin (u) cos^{\frac{4}{3}} (u) \frac{1}{5} d u

Take the constant out:

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

= \frac{1}{5} \cdot \int sin (u) cos^{\frac{4}{3}} (u) d u

Apply u-substitution

: \int - v^{\frac{4}{3}} d v

= \frac{1}{5} \cdot \int - v^{\frac{4}{3}} d v

Take the constant out:

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

= \frac{1}{5} (- \int v^{\frac{4}{3}} d v)

Apply the Power Rule

: \frac{3}{7} v^{\frac{7}{3}}

= \frac{1}{5} (- \frac{3}{7} v^{\frac{7}{3}})

Substitute back

= \frac{1}{5} (- \frac{3}{7} cos^{\frac{7}{3}} (5 x))

Simplify

\frac{1}{5} (- \frac{3}{7} cos^{\frac{7}{3}} (5 x)) : - \frac{3}{35} cos^{\frac{7}{3}} (5 x)

= - \frac{3}{35} cos^{\frac{7}{3}} (5 x)

Add a constant to the solution

= - \frac{3}{35} cos^{\frac{7}{3}} (5 x) + C

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What is the integral of sin(5x)cos^{4/3}(5x) ?
The integral of sin(5x)cos^{4/3}(5x) is -3/35 cos^{7/3}(5x)+C