# 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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##### 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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