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Substitution with the power rule worksheets

Practice indefinite u-substitution with linear and quadratic inner functions and exact positive, negative and fractional powers.

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Choose u = g(x), convert the full differential factor, integrate in u, then substitute back and include + C.

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Transform the complete differential

Set u equal to the inner polynomial, rewrite the multiplier and dx as an exact multiple of du, then use the power rule in u. Substitute the original polynomial back into the primitive and keep the same + C.

Learning objective

What this activity builds

Choose u = g(x), convert the complete differential factor, integrate in u and substitute back with + C.

Worked example

Find the indefinite integral of 2x(x^2 + 1)^3 dx.

u = x^2 + 1, du = 2x dx; integral u^3 du = u^4/4 + C = (x^2 + 1)^4/4 + C

The derivative factor converts the entire differential to du, then the power rule applies in u before back-substitution.

Teaching notes

Make the practice count

  • Group the complete multiplier with dx before replacing it by a multiple of du.
  • Keep the same integration constant through the equal antiderivative expressions.
  • State the original domain for negative and half-integer powers.

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Choose linear or quadratic inner functions and any mix of positive, negative and fractional powers.

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Math SheetNameScoreTeacherDateSubstitution with the Power RuleChoose u = g(x), convert the full differential factor, integrate in u, then substitute back and include + C.1.12·x+2·3·x2+x+112dx2.-1·x-21dx3.6·x+3·3·x2+3·x-3-2dx4.-2·-x+32dx5.-23·-x+332dx6.-4·x·x2+3-3dx7.6·-3·x+23dx8.-2·x·-2·x2+3-12dxmath-sheet.comPage 1 of 2
Math SheetNameScoreTeacherDateSubstitution with the Power RuleChoose u = g(x), convert the full differential factor, integrate in u, then substitute back and include + C.9.2·x+1·2·x2+2·x-4dx10.-23·-x-34dx11.6·x·2·x2+1-32dx12.-92·-3·x+1-2dx13.12·x+4·-3·x2-2·x1dx14.-2·x-3-3dx15.-2·x-32·2·x2+3·x-212dx16.4·2·x-12dxmath-sheet.comPage 2 of 2

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Find exact indefinite integrals of polynomial, reciprocal integer-power, and reciprocal rational-power functions while preserving the original real domain.

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