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Popular >

integral of (cos(ax)+sin(ax))^2

Solution

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

Solution

x - \frac{1}{2 a} cos (2 a x) + C

Solution steps

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

Rewrite using trig identities

= \int 1 + sin (2 x a) 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 x a) d x

\int 1 d x = x

\int sin (2 x a) d x = - \frac{1}{2 a} cos (2 a x)

= x - \frac{1}{2 a} cos (2 a x)

Add a constant to the solution

= x - \frac{1}{2 a} cos (2 a x) + C

Popular Examples

integral of (xsqrt(x+1)+(ln^2(x))/(x^2))

\int (x x + 1 + \frac{ln ^{2} ( x )}{x ^{2}}) d x

integral of (x^2+2x+1)/(x^3-16x)

\int \frac{x ^{2} + 2 x + 1}{x ^{3} - 16 x} d x

integral of (4x)/(3+2x^2)

\int \frac{4 x}{3 + 2 x ^{2}} d x

integral of 2^x+3

\int 2^{x} + 3 d x

integral of (xe^{-3x})

\int (x e^{- 3 x}) d x

Frequently Asked Questions (FAQ)

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