What is the integral of 4sin^3(2x)cos(2x) ?

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

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integral of 4sin^3(2x)cos(2x)

\int 4 sin^{3} (2 x) cos (2 x) d x

\frac{1}{2} sin^{4} (2 x) + C

Solution steps

\int 4 sin^{3} (2 x) cos (2 x) d x

Take the constant out:

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

= 4 \cdot \int sin^{3} (2 x) cos (2 x) d x

Apply u-substitution

Take the constant out:

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

Apply the Power Rule

: \frac{3}{4} u^{\frac{4}{3}}

= 4 \cdot \frac{1}{6} \cdot \frac{3}{4} u^{\frac{4}{3}}

Substitute back

u = sin^{3} (2 x)

= 4 \cdot \frac{1}{6} \cdot \frac{3}{4} (sin^{3} (2 x))^{\frac{4}{3}}

Simplify

4 \cdot \frac{1}{6} \cdot \frac{3}{4} (sin^{3} (2 x))^{\frac{4}{3}} : \frac{1}{2} sin^{4} (2 x)

= \frac{1}{2} sin^{4} (2 x)

Add a constant to the solution

= \frac{1}{2} sin^{4} (2 x) + C

\int x cos (\frac{x ^{2} + y ^{2}}{2}) d x

\int (\frac{5}{x} + 4^{x}) d x

\int \frac{3}{x ^{2} + 81} d x

\frac{cos ( x )}{x ^{2}}

\int \frac{\frac{1}{x}}{x ^{2} + 4} d x

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