What is the integral of sin(x)in(x) ?

The integral of sin(x)in(x) is d 1/2 (x-1/2 sin(2x))+C

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integral of sin(x)in(x)

\int sin (x) d sin (x) d x

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

Solution steps

\int sin (x) d sin (x) d x

Take the constant out:

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

= d \cdot \int sin (x) sin (x) d x

Simplify

sin (x) sin (x) : sin^{2} (x)

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

Rewrite using trig identities

= d \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

= d \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

= d \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)

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

Add a constant to the solution

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

x \to 3 lim (x^{3} - x^{2} + 15)

y = 0, y = cos (x), [0, \frac{3 π}{2}]

x \to 0 lim (\frac{- 2 x}{x - 3})

\int 10 x (x^{2} + 3)^{4} d x

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