Indefinite Integral Help?

Question

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!!! :)))

Answer 1

1).

Mathematical formula: x^a/x^b = x^a-b

Integrals of rational functions:

Comments

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.

Answer 2

2).

Multiply nominator and denominator by 3.

Let

Integrals of rational functions:

But u = x³ + 3x² - 5

Answer 3

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

Multiply nominator and denominator by 2.

Integrals of rational functions:

Answer 4

4).

Now take

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

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

But

Answer 5

5).

Now let

Mathematical formula: (a - b)² = a²- 2ab + b² and a^maⁿ = a^2m+n

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

Answer 6

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