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

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

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integral of sin^2(5x)

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

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

Solution steps

\int sin^{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

\frac{\partial}{\partial x} (5 x^{2} + y^{2})

\frac{d y}{d x} = \frac{y ^{2} - 1}{x ^{2} - 1}, y (5) = 5

\frac{d}{d x} ((3 x - 2)^{5})

\int 4 e^{x} 8 + e^{x} d x

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