126k views
7 votes
Derivative of an integral on x to x^2 of sin t

User Lafferc
by
7.3k points

1 Answer

9 votes

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)}

User DavidRguez
by
7.2k points