What is the integral of cos^4(ax) ?

The integral of cos^4(ax) is 1/a (cos^3(ax)sin(ax)+3/8 (ax-1/4 sin(4ax)))+C

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integral of cos^4(ax)

\int cos^{4} (a x) d x

\frac{1}{a} (cos^{3} (a x) sin (a x) + \frac{3}{8} (a x - \frac{1}{4} sin (4 a x))) + C

Solution steps

\int cos^{4} (a x) d x

Apply u-substitution

= \int \frac{cos ^{4} ( u )}{a} d u

Take the constant out:

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

= \frac{1}{a} \cdot \int cos^{4} (u) d u

Simplify

cos^{4} (u) : cos^{3} (u) cos (u)

= \frac{1}{a} \cdot \int cos^{3} (u) cos (u) d u

Apply Integration By Parts

= \frac{1}{a} (cos^{3} (u) sin (u) - \int - 3 cos^{2} (u) sin^{2} (u) d u)

\int - 3 cos^{2} (u) sin^{2} (u) d u = - \frac{3}{8} (u - \frac{1}{4} sin (4 u))

= \frac{1}{a} (cos^{3} (u) sin (u) - (- \frac{3}{8} (u - \frac{1}{4} sin (4 u))))

Substitute back

u = a x

= \frac{1}{a} (cos^{3} (a x) sin (a x) - (- \frac{3}{8} (a x - \frac{1}{4} sin (4 a x))))

Simplify

= \frac{1}{a} (cos^{3} (a x) sin (a x) + \frac{3}{8} (a x - \frac{1}{4} sin (4 a x)))

Add a constant to the solution

= \frac{1}{a} (cos^{3} (a x) sin (a x) + \frac{3}{8} (a x - \frac{1}{4} sin (4 a x))) + C

What is the integral of cos^4(ax) ?
The integral of cos^4(ax) is 1/a (cos^3(ax)sin(ax)+3/8 (ax-1/4 sin(4ax)))+C