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# Integration by Parts and Integration by Substitution (30 Problems)

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

1) &#8747; x ln x dx, u=ln x, du=xdx
2) &#8747; theta sec^2(theta) dtheta, u=theta, du=sec^2(theta) dtheta

Evaluate the integral

1) &#8747; x cos 5x dx
2) &#8747; (x)(e)^(-x) dx
3) &#8747; re^(r/2) dr
4) &#8747; t sin 2t dt
5) &#8747; x^2 sin pix dx
6) &#8747; x^2 cos mx dx
7) &#8747; ln(2x+1) dx
8) &#8747; sin^-1 (x) dx
9) &#8747; arctan 4t dt
10) &#8747; (ln x)^2 dx
11) &#8747; t^3 e^t dt
12) &#8747; e^(2theta) (sin) [3(theta)] [d(theta)]
13) &#8747; e^(-theta) (cos) [(2theta)] [d(theta)]
14) &#8747; y sinh y dy
15) &#8747; y cosh ay dy
16) &#8747; (from 0 to pi on top) t sin 3t dt
17) &#8747; (from 0 to 1 on top) (x^2 + 1) (e^(-x)) dx
18) &#8747; (from 1 to 2 on top) [ln x / x^2] dx
19) &#8747; (from 1 to 4 on top) SQRT(t) * (ln t) dt
20) &#8747; (from 0 to 1 on top) [ (y) / (e^(2y))] dy
21) &#8747; (pi/4 to pi/2 on top) (x) (csc^2)(x) dx
22) &#8747; (from 0 to ½ on top) cos^(-1)(x) dx
23) &#8747; cos x ln(sin x) dx
24) &#8747; (from1 to SQRT(3) on top) arctan(1/x) dx
25) &#8747; cos(ln x) dx
26) &#8747; (from 0 to 1 on top) [r^(3) / (SQRT(4 + r^2))] dr
27) &#8747; (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) &#8747; sin SQRT(x) dx
29) &#8747; x^5 e^x^2 dx
30) &#8747; (from 1 to 4 on top) e^(SQRT(x)) dx
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