Calculus practice

Inverse Trigonometric Integration worksheets

Practice six inverse trigonometric antiderivative rules with exact coefficients, scaled affine arguments and strict real domains.

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Use the requested inverse trigonometric function, adjust for its scaled affine derivative pattern, and include + C. Mathematically equivalent antiderivatives are accepted.

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Reverse a scaled inverse derivative

Use the named arc function, match its radical or sum-of-squares kernel, then divide by the exact signed slope and radius multiplier.

Principal ranges: arcsin [-pi/2, pi/2], arccos [0, pi], arctan (-pi/2, pi/2), arccot (0, pi), arcsec [0, pi] excluding pi/2, and arccsc [-pi/2, pi/2] excluding 0.

Learning objective

What this activity builds

Recognize the requested inverse trigonometric primitive, divide by its exact signed scaled derivative coefficient, and state the original open domain.

Worked example

Use arctan to integrate 2/(9 + (2x - 1)^2) dx.

(1/3)arctan((2x - 1)/3) + C

The derivative pattern contributes 2·3 = 6, so divide the original coefficient 2 by 6.

Teaching notes

Make the practice count

Read the requested arc-function cue before matching the kernel.
For arctan, arccot, arcsec and arccsc, include the radius in the derivative multiplier.
Keep |g(x)| in arcsec and arccsc kernels and use strict endpoint exclusions.

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Choose 8, 10 or 12 problems and the inverse trigonometric answer functions to practice.

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Practice six direct trigonometric antiderivative patterns with exact constant coefficients and unit or affine arguments in radians.

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