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Substitution in definite integrals worksheets

Transform definite integrals with linear and quadratic substitutions, new bounds and exact optional evaluation.

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Choose u = g(x), transform both bounds and the differential, then write the equivalent integral in u.

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Transform the whole definite integral

Set u equal to the inner polynomial, rewrite its derivative factor as a multiple of du, and transform both endpoints. The original order of the bounds stays intact even when the substitution decreases or maps both endpoints to the same value.

Learning objective

What this activity builds

Choose u = g(x), transform both bounds and the differential, and evaluate exactly when requested.

Worked example

Transform the integral from 0 to 2 of 2x(x² + 1)³ dx.

u = x² + 1, du = 2x dx; bounds 1 to 5; integral from 1 to 5 of u³ du

Replace the complete inner function by u, use its derivative for du, and transform each original endpoint before integrating.

Teaching notes

Make the practice count

  • Transform the lower and upper bounds before working in u.
  • Keep reversed or equal transformed bounds in their original order.
  • Check the complete interval for reciprocal, negative-power and square-root domains.

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Math SheetNameScoreTeacherDateSubstitution in Definite IntegralsChoose u = g(x), transform both bounds and the differential, then write the equivalent definite integral in u.1.-323·x+32·sinx2+x+2dx2.-2-3-3·-3·x+1-2dx3.01-3·2·x+21dx4.2-3-2·x·2·x2+3dx5.13-18·x·cos-3·x2+2dx6.-2-33·e-3·x+2dx7.03-12·1x+3dx8.2-1-3·x+1·-3·x2+2·x-22dxmath-sheet.comPage 1 of 2
Math SheetNameScoreTeacherDateSubstitution in Definite IntegralsChoose u = g(x), transform both bounds and the differential, then write the equivalent definite integral in u.9.-136·x-32·cos2·x2-x+2dx10.-2-3-23·1-x+3dx11.0-36·x+3·e-3·x2-3·x+3dx12.-3-2-92·-3·x+3-3dx13.03-23·sin-x+2dx14.32-2·x-12·2·x2+x+1dx15.23-9·x-92·e-3·x2-3·x-1dx16.-1-3-92·1-3·x-1dxmath-sheet.comPage 2 of 2

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