Substitution in definite integrals

Substitute as for an indefinite integral, but also change the limits: replace a and b by the values the new variable takes at x = a and x = b.

With the limits changed, evaluate directly in the new variable. There is no need to return to x.

Worked example: in ∫₋₁¹ 5x⁴√(x⁵ + 1) dx put t = x⁵ + 1, dt = 5x⁴ dx. The limits become t = 0 and t = 2, and the value is [(2/3)t^(3/2)] from 0 to 2 = 4√2/3.

Worked example: in ∫₀¹ tan⁻¹x/(1 + x²) dx put t = tan⁻¹x. The limits become 0 and π/4, and the value is [t²/2] from 0 to π/4 = π²/32.

Either route is correct: change the limits and stay in t, or find the antiderivative back in x and use the original limits. Mixing the two (new variable, old limits) is the usual error.