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    Indefinite Integral

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    Evaluate: Integral Sign(top 4, bottom 0) dx
    _____
    (2x + 1)^3/2

    Evaluate: Integral Sign (top 4, bottom 1) (square root x + 1/square root x) dx

    Please explain every step.

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    https://brainmass.com/math/integrals/indefinite-integral-typified-362790

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    1. Evaluate:

    Integral_(x = 0 to x = 4) { 1/(2x + 1)^3/2 } dx

    In a more general form:

    Integral_(x = a to x = b) { f(x) } dx, where
    a = 0, b = 4; f(x) = 1/(2x + 1)^3/2

    To evaluate this definite integral, we first find the anti-derivative of f(x):

    Integral { f(x) } dx = F(x) + C

    Then the value of the definite integral is given by:

    F(b) - F(a)

    That is, we find the anti-derivative and substitute the lower and upper limits in place of x and find the difference. Note that this is actually:

    (F(b) + C) - (F(a) + C)

    But C being a constant, this gives:

    F(b) - F(a)

    So the first thing to do is to find the anti-derivative of f(x) = 1/(2x + 1)^3/2:

    Integral { 1/(2x + 1)^3/2 } dx

    Note that the term in the denominator is (2x + 1)^3/2.
    We know how to integrate 1/x^3/2. We can write it as x^(-3/2) and use the formula for anti-derivative of x^n.
    But here we have 2x + 1 instead of x. Therefore, we must reduce it to the other form by using substitution:

    Let u = 2x + 1

    Then integral is:

    Integral { 1/u^3/2 } dx

    But here we're integrating with respect to x, whereas the integrand (the function inside the integral) is in ...

    Solution Summary

    Indefinite Integral is typified are examined.

    $2.19