Inverse Trigonometry Integrals

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Presentation transcript:

Inverse Trigonometry Integrals

Derivative and Antiderivatives that Deal with the Inverse Trigonometry We know the following to be true: This shows the following indefinite integral: But, what if the value in the square root is not 1? Can we still use this antiderivative?

Derivative and Antiderivatives that Deal with the Inverse Trigonometry Investigate the following: This shows the following indefinite integral: Now investigate arccos(x).

Derivative and Antiderivatives that Deal with the Inverse Trigonometry Investigate the following: This shows the following indefinite integral: This only differs by a minus sign from arcsin(x). It will be omitted from our list.

Derivative and Antiderivatives that Deal with the Inverse Trigonometry Investigate the following: This shows the following indefinite integral: Arccot(x) will only differs by a minus sign from this. It will be omitted from our list.

Integrals Involving Inverse Trigonometric Functions If u(x) is a differentiable function and a > 0, then Arcsec(x) is challenging to prove due to sign changes.

Rewrite the integral to resemble the Rule Example 1 Evaluate: Rewrite the integral to resemble the Rule Use the Rule

Rewrite the integral to resemble the Rule Example 2 Evaluate: Rewrite the integral to resemble the Rule Still missing things…

Manipulate the Numerator so it contains the derivative of the base. Example 2 Evaluate: Manipulate the Numerator so it contains the derivative of the base. Complete the square.

1980 AB Free Response 4