Derivative of integral? Please help me

Question

Find d/dt. ∫⁴_t^2 √(e^x + sin(5x²)) dx . The e to the x plus sin(5x²) is under the square root. The dx is not under sqrrt.

Answer 1

Hi, Sam:

The rule is really simple - just replace x (inside the integral) with t.

However, the upper limit of the integral must be t or a function of while the lower limit must be a constant, say 4.

In this case, we want to differentiate with respect to t but the integral goes from t² (at the lower limit) to 4 (at the upper limit).

Therefore, we must reverse the two limits which is the same as putting a minus sign in front of the integral.

Now, we can replace x with t, inside the integral, i.e., before the dx.

Since we have a function of a function, we must also apply the Chain Rule and multiply by the derivative of the inner function, t^2.

Therefore, the required derivative is: - sqrt( e^t + sin(5t²)) . (2t) [the 2t is the derivative of t²]