What is the integral of cos^2(2(x+4)) ?

The integral of cos^2(2(x+4)) is x-1/8 (2(2x+8)-sin(2(2x+8)))+C

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integral of cos^2(2(x+4))

\int cos^{2} (2 (x + 4)) d x

x - \frac{1}{8} (2 (2 x + 8) - sin (2 (2 x + 8))) + C

Solution steps

\int cos^{2} (2 (x + 4)) d x

Expand

2 (x + 4) : 2 x + 8

= \int cos^{2} (2 x + 8) d x

Rewrite using trig identities

= \int 1 - sin^{2} (2 x + 8) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 1 d x - \int sin^{2} (2 x + 8) d x

\int 1 d x = x

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

= x - \frac{1}{8} (2 (2 x + 8) - sin (2 (2 x + 8)))

Add a constant to the solution

= x - \frac{1}{8} (2 (2 x + 8) - sin (2 (2 x + 8))) + C

\int x^{3} + 1 d x

\int (sec (x) + 3)^{2} d x

\int (2 x e^{2 x}) d x

\int e^{7 x} d x

y^{^{''}} + 2 y^{^{'}} + 1 y = 1, y (1) = 1, y^{^{'}} (1) = 0

What is the integral of cos^2(2(x+4)) ?
The integral of cos^2(2(x+4)) is x-1/8 (2(2x+8)-sin(2(2x+8)))+C