Evaluate the following integrals: 1) integral (2-sqrt(x))2 dx 2) integral (t6-t2)/(t4) dt 3) integral (x2+1)/(sqrt(x)) dx

Here are the solutions for the first 9 questions.

1) Evaluate $\int (2-\sqrt{x})^2 dx$

Step 1: Expand the integrand.

$$(2-\sqrt{x})^2 = 4 - 4\sqrt{x} + x = 4 - 4x^{1/2} + x$$

Step 2: Integrate each term using the power rule $\int x^n dx = \frac{x^{n+1}}{n+1} + C$.

$$\int (4 - 4x^{1/2} + x) dx = 4x - 4\frac{x^{3/2}}{3/2} + \frac{x^2}{2} + C$$

Step 3: Simplify the expression.

$$= 4x - \frac{8}{3}x^{3/2} + \frac{1}{2}x^2 + C$$

The final answer is $\boxed{\frac{1}{2}x^2 - \frac{8}{3}x^{3/2} + 4x + C}$.

2) Evaluate $\int \frac{t^6-t^2}{t^4} dt$

Step 1: Simplify the integrand by dividing each term in the numerator by $t^4$.

$$\frac{t^6-t^2}{t^4} = \frac{t^6}{t^4} - \frac{t^2}{t^4} = t^2 - t^{-2}$$

Step 2: Integrate each term using the power rule.

$$\int (t^2 - t^{-2}) dt = \frac{t^{2+1}}{2+1} - \frac{t^{-2+1}}{-2+1} + C$$

Step 3: Simplify the expression.

$$= \frac{t^3}{3} - \frac{t^{-1}}{-1} + C = \frac{t^3}{3} + \frac{1}{t} + C$$

The final answer is $\boxed{\frac{t^3}{3} + \frac{1}{t} + C}$.

3) Evaluate $\int \frac{x^2+1}{\sqrt{x}} dx$

Step 1: Rewrite the integrand by dividing each term in the numerator by $\sqrt{x} = x^{1/2}$.

$$\frac{x^2+1}{\sqrt{x}} = \frac{x^2}{x^{1/2}} + \frac{1}{x^{1/2}} = x^{2-1/2} + x^{-1/2} = x^{3/2} + x^{-1/2}$$

Step 2: Integrate each term using the power rule.

$$\int (x^{3/2} + x^{-1/2}) dx = \frac{x^{3/2+1}}{3/2+1} + \frac{x^{-1/2+1}}{-1/2+1} + C$$

Step 3: Simplify the expression.

$$= \frac{x^{5/2}}{5/2} + \frac{x^{1/2}}{1/2} + C = \frac{2}{5}x^{5/2} + 2x^{1/2} + C$$

The final answer is $\boxed{\frac{2}{5}x^{5/2} + 2\sqrt{x} + C}$.

4) Evaluate $\int 4(\sqrt{u} + \sqrt[3]{u}) du$

Step 1: Rewrite the terms with fractional exponents.

$$4(\sqrt{u} + \sqrt[3]{u}) = 4(u^{1/2} + u^{1/3})$$

Step 2: Integrate each term using the power rule.

$$\int 4(u^{1/2} + u^{1/3}) du = 4 \left( \frac{u^{1/2+1}}{1/2+1} + \frac{u^{1/3+1}}{1/3+1} \right) + C$$

Step 3: Simplify the expression.

$$= 4 \left( \frac{u^{3/2}}{3/2} + \frac{u^{4/3}}{4/3} \right) + C = 4 \left( \frac{2}{3}u^{3/2} + \frac{3}{4}u^{4/3} \right) + C$$ $$= \frac{8}{3}u^{3/2} + 3u^{4/3} + C$$

The final answer is $\boxed{\frac{8}{3}u^{3/2} + 3u^{4/3} + C}$.

5) Evaluate $\int \frac{5}{\sqrt{x}} dx$

Step 1: Rewrite the integrand with a fractional exponent.

$$\frac{5}{\sqrt{x}} = 5x^{-1/2}$$

Step 2: Integrate using the power rule.

$$\int 5x^{-1/2} dx = 5 \frac{x^{-1/2+1}}{-1/2+1} + C$$

Step 3: Simplify the expression.

$$= 5 \frac{x^{1/2}}{1/2} + C = 10x^{1/2} + C = 10\sqrt{x} + C$$

The final answer is $\boxed{10\sqrt{x} + C}$.

6) Evaluate $\int [\sqrt{t} - \frac{2}{\sqrt{t}}] dt$

Step 1: Rewrite the terms with fractional exponents.

$$\sqrt{t} - \frac{2}{\sqrt{t}} = t^{1/2} - 2t^{-1/2}$$

Step 2: Integrate each term using the power rule.

$$\int (t^{1/2} - 2t^{-1/2}) dt = \frac{t^{1/2+1}}{1/2+1} - 2\frac{t^{-1/2+1}}{-1/2+1} + C$$

Step 3: Simplify the expression.

$$= \frac{t^{3/2}}{3/2} - 2\frac{t^{1/2}}{1/2} + C = \frac{2}{3}t^{3/2} - 4t^{1/2} + C$$

The final answer is $\boxed{\frac{2}{3}t^{3/2} - 4\sqrt{t} + C}$.

7) Evaluate $\int \frac{x^4-1}{x^2+1} dx$

Step 1: Factor the numerator using the difference of squares formula $a^2-b^2 = (a-b)(a+b)$.

$$x^4-1 = (x^2)^2 - 1^2 = (x^2-1)(x^2+1)$$

Step 2: Simplify the integrand.

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