# Indefinite Integral Help?

Can you help show how to solve these??

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

asked Feb 19, 2013 in CALCULUS

1).

Mathematical formula: xa/xb = xa-b

Integrals of rational functions:

Solution

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).

Multiply nominator and denominator by 3.

Let

Integrals of rational functions:

But u = x3 + 3x2 - 5

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

Multiply nominator and denominator by 2.

Integrals of rational functions:

4).

Now take

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

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

But

5).

Now let

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

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

6).

Consider:

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

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

But t2 = x + 3

Again consider:

Take u2 = x + 3 then 2udu = dx

But u2 = x + 3 then u = √(x+3)

Therefore