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

The integral of sin(x^2-5) is -sin(5)sqrt(pi/2)\C(sqrt(2/pi)x)+cos(5)sqrt(pi/2)\S(sqrt(2/pi)x)+C

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integral of sin(x^2-5)

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

- sin (5) \frac{π}{2} C (\frac{2}{π} x) + cos (5) \frac{π}{2} § (\frac{2}{π} x) + C

Solution steps

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

Rewrite using trig identities

= \int - cos (x^{2}) sin (5) + cos (5) sin (x^{2}) 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 (x^{2}) sin (5) d x + \int cos (5) sin (x^{2}) d x

\int cos (x^{2}) sin (5) d x = sin (5) \frac{π}{2} C (\frac{2}{π} x)

\int cos (5) sin (x^{2}) d x = cos (5) \frac{π}{2} § (\frac{2}{π} x)

= - sin (5) \frac{π}{2} C (\frac{2}{π} x) + cos (5) \frac{π}{2} § (\frac{2}{π} x)

Add a constant to the solution

= - sin (5) \frac{π}{2} C (\frac{2}{π} x) + cos (5) \frac{π}{2} § (\frac{2}{π} x) + C

What is the integral of sin(x^2-5) ?
The integral of sin(x^2-5) is -sin(5)sqrt(pi/2)\C(sqrt(2/pi)x)+cos(5)sqrt(pi/2)\S(sqrt(2/pi)x)+C