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Substitution with inverse trigonometric functions worksheets

Practice indefinite u-substitution into all six inverse-trigonometric integration patterns with polynomial, logarithmic and trigonometric inner functions.

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

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Transform the complete differential

Set u equal to the complete polynomial, logarithmic or trigonometric inner function. The numerator becomes k du before matching the inverse-trigonometric pattern.

Learning objective

What this activity builds

Choose u = g(x), convert the complete differential, match the exact inverse-trigonometric kernel, and substitute back while retaining every original domain restriction.

Worked example

Find the indefinite integral after identifying u = g(x).

Convert k g'(x) dx to k du, integrate the matching inverse-trigonometric kernel, and replace u by g(x).

The primitive coefficient is k/(s m); the inner derivative has already been absorbed into du.

Teaching notes

Make the practice count

  • Keep the complete inner derivative with dx.
  • Do not divide by the inner slope a second time.
  • Retain logarithm and trigonometric pole restrictions together with the inverse-kernel inequality.

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Choose 8 or 16 problems, all six inverse answers, polynomial/logarithmic/trigonometric inners, and whether to show u = g(x).

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Math SheetNameScoreTeacherDateSubstitution with Inverse Trigonometric FunctionsChoose u = g(x), convert the complete differential, integrate the inverse-trigonometric pattern in u, then substitute back and include + C.1. Use arccot:cos-2-x9+2·sin-2-x-22dxu =2·sin-2-x-22. Use arcsec:41-2·x·1-2·x2-4dxu =1-2·x3. Use arccsc:4-3-2·x-2·ln-3-2·x-1·-2·ln-3-2·x-12-9dxu =-2·ln-3-2·x-14. Use arccos:-6·sin2-2·x4-cos2-2·x+02dxu =cos2-2·x+05. Use arctan:12+3·x1+-1-x-3·x22dxu =-1-x-3·x26. Use arcsin:43-3-x9-2·ln-3-x-12dxu =2·ln-3-x-17. Use arcsin:2+12·x-12·x21--2-x-3·x2+2·x32dxu =-2-x-3·x2+2·x38. Use arccsc:-32·sec-2+3·x2tan-2+3·x-1·tan-2+3·x-12-16dxu =tan-2+3·x-1Radians; arccot is in (0, pi). Retain every original open-domain restriction.math-sheet.comPage 1 of 1

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