What is the integral of sin^2(x+1)-2 ?

The integral of sin^2(x+1)-2 is -3/2 x-1/4 sin(2(x+1))-1/2+C

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integral of sin^2(x+1)-2

\int sin^{2} (x + 1) - 2 d x

- \frac{3}{2} x - \frac{1}{4} sin (2 (x + 1)) - \frac{1}{2} + C

Solution steps

\int sin^{2} (x + 1) - 2 d x

Rewrite using trig identities

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

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

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

\int 1 d x = x

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

Simplify

- \frac{1}{2} (x + 1 + \frac{1}{2} sin (2 (x + 1))) - x : - \frac{3}{2} x - \frac{1}{4} sin (2 (x + 1)) - \frac{1}{2}

= - \frac{3}{2} x - \frac{1}{4} sin (2 (x + 1)) - \frac{1}{2}

Add a constant to the solution

= - \frac{3}{2} x - \frac{1}{4} sin (2 (x + 1)) - \frac{1}{2} + C

\int - 2 x^{2} + x d x

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\int csc^{6} (x) cot (x) d x

\int \frac{1}{2 t - 1} d t

What is the integral of sin^2(x+1)-2 ?
The integral of sin^2(x+1)-2 is -3/2 x-1/4 sin(2(x+1))-1/2+C