What is the integral of cos^4(5x+4) ?

The integral of cos^4(5x+4) is 1/5 (cos^3(5x+4)sin(5x+4)+3/8 (5x+4-1/4 sin(4(5x+4))))+C

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integral of cos^4(5x+4)

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

\frac{1}{5} (cos^{3} (5 x + 4) sin (5 x + 4) + \frac{3}{8} (5 x + 4 - \frac{1}{4} sin (4 (5 x + 4)))) + C

Solution steps

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

Apply u-substitution

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

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

Simplify

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

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

Apply Integration By Parts

: cos^{3} (u) sin (u) - \int - 3 cos^{2} (u) sin^{2} (u) d u

= \frac{1}{5} (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}{5} (cos^{3} (u) sin (u) - (- \frac{3}{8} (u - \frac{1}{4} sin (4 u))))

Substitute back

u = 5 x + 4

= \frac{1}{5} (cos^{3} (5 x + 4) sin (5 x + 4) - (- \frac{3}{8} (5 x + 4 - \frac{1}{4} sin (4 (5 x + 4)))))

Simplify

= \frac{1}{5} (cos^{3} (5 x + 4) sin (5 x + 4) + \frac{3}{8} (5 x + 4 - \frac{1}{4} sin (4 (5 x + 4))))

Add a constant to the solution

= \frac{1}{5} (cos^{3} (5 x + 4) sin (5 x + 4) + \frac{3}{8} (5 x + 4 - \frac{1}{4} sin (4 (5 x + 4)))) + C

f (t) = 2 (1 - e^{- 5 t})

\frac{d}{d x} (ln^{2} (2) \cdot 2^{x})

y = ln (x - 4)

\int π (\frac{9}{x ^{2} + 9})^{2} d x

4 x y \frac{d y}{d x} = 1

What is the integral of cos^4(5x+4) ?
The integral of cos^4(5x+4) is 1/5 (cos^3(5x+4)sin(5x+4)+3/8 (5x+4-1/4 sin(4(5x+4))))+C