F(x) = cos(x) • the integral from 2 to x² + 1 of

Question

e^(u² +5)du

Find F'(x).

When i did this, i got:
-2xsin(x)e^((x²+1)² + 5)

But my teacher got:
-sin(x) • the integral from x² + 1 of e^(u² +5)du + 2xcos(x)e^((x²+1)² + 5)

Do you know why the integral is in his answer? I'm not sure where I went wrong. If you could help, I would greatly appreciate it. Thanks!!

Answer 1

Leibnitz's Rule explains how to take the derivative of an integral. Take a google for it, or consult your textbook.

Basically, you have a product here. cos(x) * Integral f(x)

d/dx of the product is

-sin(x) * Integral + cos(x) * d/dx(Integral(f))

d/dx(Integral) = Integral(df/dx) = f, evaluated at the limits of integration.