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Easy: (ax + b)ⁿ
Substitution undoes the chain rule. Let u be the inside expression, find du = u′ dx, and rewrite the whole integral in u. Integrate with the power rule, then put the original expression back in place of u and add + C. If du is off by a constant factor, divide by that factor.
Integrate using substitution.
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Pick the inside of a composition whose derivative also appears (up to a constant). In ∫2x(x² + 1)³ dx, u = x² + 1 and du = 2x dx.
Adjust it. For ∫(3x + 1)⁴ dx, du = 3 dx, so dx = du/3 and the integral is (1/3)∫u⁴ du = (3x + 1)⁵/15 + C.
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