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    Trigonometric Integrals and Integration by Substitution

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    Evaluate the integral

    1) ∫ (sin^3 (x)) (cos^2 (x)) dx
    2) ∫ ( sin^4 (x)) (cos^5 (x)) dx
    3) ∫ ( sin^6 (x)) (cos^3 (x)) dx
    4) ∫ ( sin^3 (mx)) dx
    5) ∫ (from 0 to pi/2 on top) (cos^2 (theta)) dtheta
    6) ∫ (from 0 to pi/2 on top) (sin^2 (2theta)) dtheta
    7) ∫ (from 0 to pi on top) (sin^4 (3t)) dt
    8) ∫ (from 0 to pi on top) (cos^6 (theta)) dtheta
    9) ∫ ( 1 + cos theta)^2 dtheta
    10) ∫ (x) (cos^2 (x)) dx
    11) ∫ ( sin^3 (x)) (SQRT(cos (x)) dx
    12) ∫ (cos (theta) (cos^5(sin(theta))) dtheta
    13) ∫ (cos^2 (x)) (tan^3 (x)) dx
    14) ∫ ( cot^5 (theta)) (sin^4 (theta)) dtheta
    15) ∫ [( 1 - sinx) / (cosx) ]dx
    16) ∫ ( sin 2x) (cos^2 (x)) dx
    17) ∫ ( sec^2 (x)) (tan (x)) dx
    18) ∫ ( tan^2(x) dx
    19) ∫ ( sec^6(t)) dt
    20) ∫ (from 0 to pi/4 on top) (sin^4 (theta)) (tans^4 (theta)) dtheta
    21) ∫ (from 0 to pi/3 on top) (tan^5 (x)) (sec^4 (x)) dx
    22) ∫ ( tan^3 (2x)) (sec^5 (2x)) dx
    23) ∫ ( tan^3 (x)) (sec (x)) dx
    24) ∫ (from 0 to pi/3 on top) (tan^5 (x)) (sec^6 (x)) dx
    25) ∫ ( tan^5 (x)) dx
    26) ∫ ( tan^3 (theta)) / (cos^4 (theta)) dtheta
    27) ∫ ( tan^2 (x)) (sec (x)) dx
    28) ∫ ( cot^3 (alpha)) (csc^3 (alpha)) dalpha
    29) ∫ ( csc^4 (x)) (cot^6 (x)) dx
    30) ∫ ( csc (x)) dx
    31) ∫ ( sin (5x)) (sin (2x)) dx
    32) ∫ [( dx) / ((cos (x)) - 1)] dx
    33) ∫ [( 1 - tan^2 (x)) / (sec^2 (x))] dx
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    https://brainmass.com/math/integrals/trigonometric-integrals-integration-substitution-63472

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    Solution Summary

    Thirty-three trigonometric integrals are found, mostly using integration by substitution. The solution is detailed and well presented. The response was given a rating of "5/5" by the student who originally posted the question. [Editor's Note : This is the most comprehensive solution for trigonometric integrals that I have seen.]

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