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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) ∫ 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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    https://brainmass.com/math/integrals/integration-by-parts-and-integration-by-substitution-30-problems-63471

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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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