What is the integral of sin(x)sin(x+1) ?

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

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integral of sin(x)sin(x+1)

\int sin (x) sin (x + 1) d x

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

Solution steps

\int sin (x) sin (x + 1) d x

Rewrite using trig identities

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

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

\int cos (1) d x = cos (1) x

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

Add a constant to the solution

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

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