What is the integral of 7cos^4(x) ?

The integral of 7cos^4(x) is 7(cos^3(x)sin(x)+3/8 (x-1/4 sin(4x)))+C

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integral of 7cos^4(x)

\int 7 cos^{4} (x) d x

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

Solution steps

\int 7 cos^{4} (x) d x

Take the constant out:

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

= 7 \cdot \int cos^{4} (x) d x

Simplify

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

= 7 \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

= 7 (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))

= 7 (cos^{3} (x) sin (x) - (- \frac{3}{8} (x - \frac{1}{4} sin (4 x))))

Simplify

= 7 (cos^{3} (x) sin (x) + \frac{3}{8} (x - \frac{1}{4} sin (4 x)))

Add a constant to the solution

= 7 (cos^{3} (x) sin (x) + \frac{3}{8} (x - \frac{1}{4} sin (4 x))) + C

\frac{d y}{d t} + e^{t} y = 20 e^{t}

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

\frac{1}{s ( s ^{2} + 3 s + 2 )}

\frac{\partial}{\partial x} (x y e^{4 x - 2 z})

\frac{d y}{d x} + 9 y = e^{8 x}

What is the integral of 7cos^4(x) ?
The integral of 7cos^4(x) is 7(cos^3(x)sin(x)+3/8 (x-1/4 sin(4x)))+C