What is the integral of 8cos^3(2θ)sin(2θ) ?

The integral of 8cos^3(2θ)sin(2θ) is -cos^4(2θ)+C

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integral of 8cos^3(2θ)sin(2θ)

\int 8 cos^{3} (2 θ) sin (2 θ) d θ

- cos^{4} (2 θ) + C

Solution steps

\int 8 cos^{3} (2 θ) sin (2 θ) d θ

Take the constant out:

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

= 8 \cdot \int cos^{3} (2 θ) sin (2 θ) d θ

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}}

= 8 (- \frac{1}{6} \cdot \frac{3}{4} u^{\frac{4}{3}})

Substitute back

u = cos^{3} (2 θ)

= 8 (- \frac{1}{6} \cdot \frac{3}{4} (cos^{3} (2 θ))^{\frac{4}{3}})

Simplify

8 (- \frac{1}{6} \cdot \frac{3}{4} (cos^{3} (2 θ))^{\frac{4}{3}}) : - cos^{4} (2 θ)

= - cos^{4} (2 θ)

Add a constant to the solution

= - cos^{4} (2 θ) + C

x \to 0 lim (\frac{tan ( a x )}{x})

\frac{d}{d x} (\frac{1}{2} sin (x) cos (x))

(\frac{1}{y ^{2}} - \frac{7}{y ^{4}}) (y + 5 y^{3})

\frac{d}{d x} (\frac{1 - sin ( x )}{1 + cos ( 2 x )})

f (x) = - 1

What is the integral of 8cos^3(2θ)sin(2θ) ?
The integral of 8cos^3(2θ)sin(2θ) is -cos^4(2θ)+C