Question

Derivative of an integral on x to x^2 of sin t

Answer 1

Split the integral at a constant x = c and expand it as

`\displaystyle \int_x^(x^2) \sin(t) \, dt = \int_x^c \sin(t) \, dt + \int_c^(x^2) \sin(t) \, dt \\\\ = \int_c^(x^2) \sin(t) \, dt - \int_c^x \sin(t) \, dt`

Now apply the fundamental theorem of calculus.

`\displaystyle (d)/(dx) \int_c^(x^2) \sin(t) \, dt = \sin(x^2) (d)/(dx) x^2 = 2x \sin(x^2)`

`\displaystyle (d)/(dx) \int_c^x \sin(t) \, dt = \sin(x)`

and so

`\displaystyle (d)/(dx) \int_x^(x^2) \sin(t) \, dt = \boxed{2x \sin(x^2) - \sin(x)}`