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 TRIGONOMETRIC FUNCTIONS Trigonometric functions, which comprise one group of transcendental functions, may be differentiated and integrated in the same fashion as the other functions. We will limit our proofs to the sine, cosine, and secant functions but will list several others. Formula. PROOF: and Therefore, Formula. PROOF: Therefore, Formula. PROOF: and by the quotient rule Therefore, To this point we have considered integrals of trigonometric functions that result in functions of the sine, cosine, and tangent. Those integrals that result in functions of the cotangent, secant, and cosecant are included in the following list of elementary integrals: EXAMPLE: Evaluate SOLUTION: We need the integral in the form of We let so that but we do not have 3 dx. Therefore, we multiply the integral by 3/3 and rearrange as follows: EXAMPLE: Evaluate SOLUTION: Let so that Therefore, EXAMPLE: Evaluate SOLUTION: We use the rule for sums and write Then, in the integral let so that but we have 3 dx Hence, proper compensation has to be made as follows: The second integral with and is evaluated as follows: Then, by combining the two solutions, we have where EXAMPLE: Evaluate SOL UTION: Let so that We need 5 dx so we write EXAMPLE: Evaluate SOL UTION: Let so that We require du equal to 2 dx, so we write EXAMPLE: Evaluate SOL UTION: Let u=6x so that then, EXAMPLE: Evaluate SOL UTION: Let so that then,