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

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

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integral of cos^4(x/5)

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

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

Solution steps

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

Apply u-substitution

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

= \int cos^{4} (u) \cdot 5 d u

Take the constant out:

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

= 5 \cdot \int cos^{4} (u) d u

Simplify

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

= 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

= 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))

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

Substitute back

u = \frac{x}{5}

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

Simplify

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

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

Add a constant to the solution

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

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

y^{^{'}} = \frac{x ^{4} y ^{4} - 2 x ^{4}}{y ^{3}}

x \to \infty lim (3 x^{2} + 1)

\frac{d}{d x} (\frac{2}{3} x^{3} + x^{2} - 12 x + 17)

\frac{x ^{2} - 2 x - 8}{x + 2}

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