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Indefinite Integral Help?

+2 votes
Can you help show how to solve these??
Please kindly show the steps.

1. (x^2/3) - (x^1/4) / (x^1/2) dx
2. [x(x+2)] / (x^3 + 3x^2 - 5) dx
3. (X^3)(x^4 - 4x^2 + 4)^1/2 dx
4. [(x-1)/(x^5)]^1/2 dx
5. (x^2 + 3)^1/3 (x^5) dx
6. (x^3 + 3x^2 - 9) / (x + 3)^1/2 dx

Thanks in advance!!! :)))
asked Feb 19, 2013 in CALCULUS by dkinz Apprentice

6 Answers

+2 votes

1). image

Mathematical formula: xa/xb = xa-b

image

Integrals of rational functions: image

image

image

image

image

image

answered Feb 20, 2013 by britally Apprentice

Solution

image

The extra C, called the constant of integration, is really necessary, since after all differentiation kills off constants , which is why integration and differentiation are not exactly inverse operations of each other.

+2 votes

2). image

image

Multiply nominator and denominator by 3.

image

Let image

image

Integrals of rational functions: image

image

But u = x3 + 3x2 - 5

image

answered Feb 20, 2013 by britally Apprentice
+2 votes

3). Presumed (X^3)/(x^4 - 4x^2 + 4)^1/2 dx

image

image

image

image

Multiply nominator and denominator by 2.

image

image

image

image

image

Integrals of rational functions:image

image

image

answered Feb 20, 2013 by britally Apprentice
+2 votes

4).image

image

Now take image

The Quotient Rule: (u/v)' = [vu'-uv']/[v2]

image

image

Integrals of rational functions: ʃxndx = [xn+1/n+1]

image

But image

image

answered Feb 20, 2013 by britally Apprentice
+2 votes

5). image

image

Now let image

image

Mathematical formula: (a - b)2 = a2- 2ab + b2 and aman = a2m+n

image

image

image

image

Integrals of rational functions: ʃxndx = [xn+1/n+1]

image

image

image

image

image

answered Feb 20, 2013 by richardson Scholar
+2 votes

6). image

image

image

Consider:image

Now let t2 = x + 3 then (2t)dt = dx and x = t2 - 3.

image

Integrals of rational function: ʃxndx = [xn+1/n+1]+C

image

But t2 = x + 3

image

Again consider: image

Take u2 = x + 3 then 2udu = dx

image

But u2 = x + 3 then u = √(x+3)

image

Therefore

image

answered Feb 20, 2013 by richardson Scholar

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