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

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

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integral of 5cos^4(x)

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

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

Solution steps

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

Take the constant out:

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

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

Simplify

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

= 5 \cdot \int cos^{3} (x) cos (x) d x

Apply Integration By Parts

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

= 5 (cos^{3} (x) sin (x) - \int - 3 cos^{2} (x) sin^{2} (x) d x)

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

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

Simplify

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

Add a constant to the solution

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

\frac{\partial}{\partial z} (- 3 z)

x \cdot cos (\frac{3 π x}{2}), 0

\int_{0}^{\frac{4}{5}} \frac{1}{16 + 25 x ^{2}} d x

y = sin^{2} (x) \frac{d y}{d x}

\int (x^{3} + x^{- 4} + x^{\frac{3}{4}}) d x

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