Can someone show me how to evaluate the integral from 0 to 5 of x/(x^2 + 4) dx ?

To evaluate the integral ∫(0 to 5) of x/(x^2 + 4) dx, we can apply a technique called substitution. Here's a step-by-step explanation of how to solve this integral:

Step 1: Identify the substitution. Let's substitute u = x^2 + 4. This choice is beneficial because when we differentiate u with respect to x, we get du/dx = 2x.

Step 2: Find the differential dx in terms of du. Since we have u = x^2 + 4, differentiating both sides with respect to x gives du = (2x)dx. Rearranging this equation, we get dx = du/(2x).

Step 3: Rewrite the integral in terms of u and x. The limits of integration (0 to 5) are for x, so we need to express them in terms of u. When x = 0, u = 0^2 + 4 = 4. When x = 5, u = 5^2 + 4 = 29. So the new integral becomes ∫(4 to 29) of (x/(x^2 + 4)) dx = ∫(0 to 5) of (x/u) (du/(2x)).

Step 4: Simplify the integral. Cancel out the x terms from the numerator and denominator to get ∫(0 to 5) of (1/u) du/2.

Step 5: Evaluate the integral. Taking the integral of 1/u with respect to u gives ln|u|. Divide by 2 to obtain (1/2)ln|u|.

Step 6: Substitute back the original variable. Replace u with x^2 + 4 to get the final result, which is (1/2)ln|x^2 + 4|.

Therefore, the evaluation of the integral ∫(0 to 5) of x/(x^2 + 4) dx is equal to (1/2)ln|x^2 + 4|, with the limits of integration applied.