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Calculus practice

Logarithmic and exponential substitution worksheets

Practice indefinite u-substitution whose primitive is exponential or logarithmic, with polynomial, exponential and logarithmic inner functions.

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

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Transform the complete differential

Set u equal to the complete inner function, including any shift or scale. Rewrite the multiplier and dx as an exact multiple of du before integrating.

Learning objective

What this activity builds

Choose u = g(x), convert the complete differential, integrate an exponential or reciprocal expression in u, and substitute back with + C.

Worked example

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

u = x^2 + 1, du = 2x dx; integral e^u du = e^u + C = e^(x^2 + 1) + C

The complete derivative factor becomes du, leaving the exponential primitive in u.

Teaching notes

Make the practice count

  • Group the full multiplier with dx before replacing it by a multiple of du.
  • Use ln|u| for reciprocal primitives.
  • Keep the original conditions h(x) > 0 and g(x) != 0 where they apply.

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Choose exponential or logarithmic answers, polynomial or exponential/logarithmic inner functions, and whether to provide u.

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Math SheetNameScoreTeacherDateLogarithmic and Exponential SubstitutionChoose u = g(x), convert the complete differential, integrate in u, then substitute back and include + C.1.-43-2·xdxu =-2·x2.3·e-x+2-e-x+2+1dxu =-e-x+2+13.32·e3·x-1dxu =3·x-14.32·e-3·x-2·ee-3·x-2+1dxu =e-3·x-2+15.-12-x-3·eln-x-3-1dxu =ln-x-3-16.-8·x+22·x2-x-2dxu =2·x2-x-27.12-x+3ln-x+3+2dxu =ln-x+3+28.2·x-2·ex2-2·x+3dxu =x2-2·x+3math-sheet.comPage 1 of 1

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