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

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

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integral of 5sin^2(x)

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

\frac{5}{2} (x - \frac{1}{2} sin (2 x)) + C

Solution steps

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

Take the constant out:

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

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

Rewrite using trig identities

= 5 \cdot \int \frac{1 - cos ( 2 x )}{2} d x

Take the constant out:

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

= 5 \cdot \frac{1}{2} \cdot \int 1 - cos (2 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

= 5 \cdot \frac{1}{2} (\int 1 d x - \int cos (2 x) d x)

\int 1 d x = x

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

= 5 \cdot \frac{1}{2} (x - \frac{1}{2} sin (2 x))

Simplify

5 \cdot \frac{1}{2} (x - \frac{1}{2} sin (2 x)) : \frac{5}{2} (x - \frac{1}{2} sin (2 x))

= \frac{5}{2} (x - \frac{1}{2} sin (2 x))

Add a constant to the solution

= \frac{5}{2} (x - \frac{1}{2} sin (2 x)) + C

\frac{d}{d x} (2 x^{2} - 5 x + 4)

100

\int (x - 4)^{2} x^{2} d x

\frac{\partial}{\partial x} (- 3 sin (2 x))

\frac{d}{d x} (2 a sin (x))

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