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Indefinite Integral

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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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.