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# Trigonometric Integrals and Integration by Substitution

Evalaute the integral

1) &#8747; (sin^3 (x)) (cos^2 (x)) dx
2) &#8747; ( sin^4 (x)) (cos^5 (x)) dx
3) &#8747; ( sin^6 (x)) (cos^3 (x)) dx
4) &#8747; ( sin^3 (mx)) dx
5) &#8747; (from 0 to pi/2 on top) (cos^2 (theta)) dtheta
6) &#8747; (from 0 to pi/2 on top) (sin^2 (2theta)) dtheta
7) &#8747; (from 0 to pi on top) (sin^4 (3t)) dt
8) &#8747; (from 0 to pi on top) (cos^6 (theta)) dtheta
9) &#8747; ( 1 + cos theta)^2 dtheta
10) &#8747; (x) (cos^2 (x)) dx
11) &#8747; ( sin^3 (x)) (SQRT(cos (x)) dx
12) &#8747; (cos (theta) (cos^5(sin(theta))) dtheta
13) &#8747; (cos^2 (x)) (tan^3 (x)) dx
14) &#8747; ( cot^5 (theta)) (sin^4 (theta)) dtheta
15) &#8747; [( 1 - sinx) / (cosx) ]dx
16) &#8747; ( sin 2x) (cos^2 (x)) dx
17) &#8747; ( sec^2 (x)) (tan (x)) dx
18) &#8747; ( tan^2(x) dx
19) &#8747; ( sec^6(t)) dt
20) &#8747; (from 0 to pi/4 on top) (sin^4 (theta)) (tans^4 (theta)) dtheta
21) &#8747; (from 0 to pi/3 on top) (tan^5 (x)) (sec^4 (x)) dx
22) &#8747; ( tan^3 (2x)) (sec^5 (2x)) dx
23) &#8747; ( tan^3 (x)) (sec (x)) dx
24) &#8747; (from 0 to pi/3 on top) (tan^5 (x)) (sec^6 (x)) dx
25) &#8747; ( tan^5 (x)) dx
26) &#8747; ( tan^3 (theta)) / (cos^4 (theta)) dtheta
27) &#8747; ( tan^2 (x)) (sec (x)) dx
28) &#8747; ( cot^3 (alpha)) (csc^3 (alpha)) dalpha
29) &#8747; ( csc^4 (x)) (cot^6 (x)) dx
30) &#8747; ( csc (x)) dx
31) &#8747; ( sin (5x)) (sin (2x)) dx
32) &#8747; [( dx) / ((cos (x)) - 1)] dx
33) &#8747; [( 1 - tan^2 (x)) / (sec^2 (x))] dx
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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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