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7.3 CALCULUS WITH THE INVERSE TRIGONOMETRIC FUNCTIONS. The three previous sections introduced the ideas of one–to–one functions and inverse functions and used. those ideas to define arcsine, arctangent, and the other inverse trigonometric functions. Section 7.3 presents. the calculus of inverse trigonometric functions.
Integrals of Trigonometric Functions. ∫ sin x dx = − cos x + C. ∫ cos x dx = sin x + C. ∫ tan x dx = ln sec x + C. ∫ sec x dx = ln tan x + sec x + C. ∫ 1. sin. 2. x dx = ( x − sin x cos x ) + C.
The Rule: . arctan. OR . arctan. Putting everything together: . Use this rule when you have a fraction of the form: where the polynomial in the denominator does . not factor and the fraction is not in the correct form to turn into an . Example: evaluate . From the example above we know that .
Inverse Trigonometric Functions: Integration. Integrate functions whose antiderivatives involve inverse trigonometric functions. Use the method of completing the square to integrate a function. Review the basic integration rules involving elementary functions.
STRATEGY FOR EVALUATING R tanm(x) secn(x)dx. If the power n of secant is even (n = 2k, k 2), save one sec2(x) factor and use sec2(x) = 1+tan2(x) to express the rest of the factors in terms of tangent: tanm(x) secn(x)dx = Z tanm(x) sec2k(x)dx = =.
x INTEGRAL RULES. ∫ sin xdx = − cos x + c. ∫ cos xdx = sin x + c. ∫ sec 2 xdx = tan x + c.
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