What is the integral of cos^2(5x) ?

The integral of cos^2(5x) is 1/2 (x+1/10 sin(10x))+C

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integral of cos^2(5x)

\int cos^{2} (5 x) d x

\frac{1}{2} (x + \frac{1}{10} sin (10 x)) + C

Solution steps

\int cos^{2} (5 x) d x

Rewrite using trig identities

= \int \frac{1}{2} (1 + cos (10 x)) d x

Take the constant out:

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

= \frac{1}{2} \cdot \int 1 + cos (10 x) 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

= \frac{1}{2} (\int 1 d x + \int cos (10 x) d x)

\int 1 d x = x

\int cos (10 x) d x = \frac{1}{10} sin (10 x)

= \frac{1}{2} (x + \frac{1}{10} sin (10 x))

Add a constant to the solution

= \frac{1}{2} (x + \frac{1}{10} sin (10 x)) + C

(1 + e^{x} y + x e^{x} y) d x + (x e^{x} + 2) d y = 0

\int \frac{9 x ^{3} - 5 x ^{2} + 3 x}{x} d x

e^{0.04 t}

x \to - 3 lim (\frac{x ^{2} - 9}{x - 3})

x \to 0 lim ([\frac{sin ( x )}{x}])

What is the integral of cos^2(5x) ?
The integral of cos^2(5x) is 1/2 (x+1/10 sin(10x))+C