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Substitution with trigonometric functions worksheets

Practice indefinite u-substitution with sine, cosine, tangent, cotangent, secant and cosecant primitives across polynomial and exponential inner functions.

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

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Transform the complete differential

Set u equal to the full polynomial or exponential inner function. Group the multiplier with dx and rewrite it as an exact multiple of du.

Learning objective

What this activity builds

Choose u = g(x), convert the complete differential, integrate a trigonometric form in u, and substitute back with + C while retaining the original domain.

Worked example

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

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

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

Teaching notes

Make the practice count

  • Group the full multiplier with dx before replacing it by a multiple of du.
  • Track the negative signs for sine and cosecant-squared primitives.
  • Keep excluded zeros from the original tangent, cotangent, secant or cosecant integrand.

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Choose 4, 5 or 6 problems, polynomial or exponential inner functions, and the trigonometric answer groups to practice.

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Math SheetNameScoreTeacherDateSubstitution with Trigonometric FunctionsChoose u = g(x), convert the complete differential, integrate the trigonometric form in u, then substitute back and include + C.1.-32·e3-x·sece3-x-1·tane3-x-1dx2.43+43·x-4·x2·cos2-2·x-x2+2·x3dx3.-12·secx2dx4.-12-2·x·csc-x-2·x22dx5.-6·x·csc-3-x2·cot-3-x2dxAll angles are in radians. Respect the original integrand domain.math-sheet.comPage 1 of 1

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