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Integration by Parts and Integration by Substitution (30 Problems)

Evaluate the integral using integration by parts with the indicated choices of u and du.

1) ∫ x ln x dx, u=ln x, du=xdx
2) ∫ theta sec^2(theta) dtheta, u=theta, du=sec^2(theta) dtheta

Evaluate the integral

1) ∫ x cos 5x dx
2) ∫ (x)(e)^(-x) dx
3) ∫ re^(r/2) dr
4) ∫ t sin 2t dt
5) ∫ x^2 sin pix dx
6) ∫ x^2 cos mx dx
7) ∫ ln(2x+1) dx
8) ∫ sin^-1 (x) dx
9) ∫ arctan 4t dt
10) ∫ (ln x)^2 dx
11) ∫ t^3 e^t dt
12) ∫ e^(2theta) (sin) [3(theta)] [d(theta)]
13) ∫ e^(-theta) (cos) [(2theta)] [d(theta)]
14) ∫ y sinh y dy
15) ∫ y cosh ay dy
16) ∫ (from 0 to pi on top) t sin 3t dt
17) ∫ (from 0 to 1 on top) (x^2 + 1) (e^(-x)) dx
18) ∫ (from 1 to 2 on top) [ln x / x^2] dx
19) ∫ (from 1 to 4 on top) SQRT(t) * (ln t) dt
20) ∫ (from 0 to 1 on top) [ (y) / (e^(2y))] dy
21) ∫ (pi/4 to pi/2 on top) (x) (csc^2)(x) dx
22) ∫ (from 0 to ½ on top) cos^(-1)(x) dx
23) ∫ cos x ln(sin x) dx
24) ∫ (from1 to SQRT(3) on top) arctan(1/x) dx
25) ∫ cos(ln x) dx
26) ∫ (from 0 to 1 on top) [r^(3) / (SQRT(4 + r^2))] dr
27) ∫ (from 1 to 2 on top) x^(4) (ln x)^2 dx

First make a substitution and then use integration by parts to evaluate the integral.
28) ∫ sin SQRT(x) dx
29) ∫ x^5 e^x^2 dx
30) ∫ (from 1 to 4 on top) e^(SQRT(x)) dx
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Solution Summary

Thirty integrals are found using Integration by Parts and 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 one of the most comprehensive, high quality integration solutions that I have seen.]

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