Find the derivative of a Definite Integral

Question

Find the derivative of g(x)= ₁∫^x x²/(√(x²+4)

Answer 1

I hope this wasn't how the question was posed. If it was, boo on your teacher.

Because this is a definite integral, the only "x" here is in the integral bound. In other words, you can rewrite this as (e.g.):

g(x) = integral(1 to x) of s^2/sqrt(s^2+4)ds

The x inside the integral is a dummy variable of integration.

As a result, the answer is: dg/dx = x^2/sqrt(x^2+4)

Answer 2

Question:

Find the derivative of g(x)= ₁∫^x x²/(√(x²+4)

Restated:

d/dx g(x) = d/dx INT [x^2 / SQRT(x^2+4)] dx from 1 to x

Formula:

d/dx INT[f(t)] dt from a to x = F'(x) = f(x)

Therefore

g'(x) = x^2 / SQRT(x^2+4)