Evaluate the definite integral of 2x(x^2 + 1)^3 from x = 0 to x = 1.

Answer

The integral equals 15/4.

Explanation

Step 1 — Choose the substitution

Let u = x^2 + 1. Then du = 2x dx, which is exactly the factor sitting in front of the bracket — the signal that u-substitution is the right method here.

Step 2 — Change the limits

Because this is a definite integral, convert the limits instead of substituting back at the end:

x	u = x² + 1
0	1
1	2

Step 3 — Integrate in u

The integral becomes the integral of u^3 du from u = 1 to u = 2:

∫ u³ du = u⁴ / 4

Step 4 — Evaluate

= (2⁴ / 4) − (1⁴ / 4) = 4 − 1/4 = 15/4

Why change the limits

Substituting back to x also works, but converting the limits removes a whole step and the most common source of sign errors in exam answers.