Question: 1. Find $\int(13 - 8x + 25x^8)\,dx$
2. Find $\int\left(-14\pi + \frac{3}{x} - \frac{6}{x^4}\right)\,dx$
3. Integrate: $\int(7 - 9x^2)^2\,dx$
4. Find $\int\frac{-15x^5 + x^3 + 10x^2}{x^3} \, dx$
5. Find $\int\left(\frac{3}{x^3} + \frac{2}{\sqrt[3]{x}}\right)\,dx$
6. $\int\sqrt{x}\left(x + \frac{2}{x}\right)\,dx$
1. **Problem:** Find $\int(13 - 8x + 25x^8)\,dx$.
**Step 1:** Use the power rule for integration: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$.
**Step 2:** Integrate each term separately:
$$\int 13 \, dx = 13x + C$$
$$\int -8x \, dx = -8 \int x \, dx = -8 \frac{x^{1+1}}{1+1} = -8 \frac{x^2}{2} = -4x^2$$
$$\int 25x^8 \, dx = 25 \frac{x^{8+1}}{8+1} = 25 \frac{x^9}{9} = \frac{25}{9} x^9$$
**Step 3:** Combine all results:
$$\int(13 - 8x + 25x^8)\,dx = 13x - 4x^2 + \frac{25}{9} x^9 + C$$
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2. **Problem:** Find $\int\left(-14\pi + \frac{3}{x} - \frac{6}{x^4}\right)\,dx$.
**Step 1:** Rewrite terms with powers:
$$\frac{3}{x} = 3x^{-1}, \quad -\frac{6}{x^4} = -6x^{-4}$$
**Step 2:** Integrate each term:
$$\int -14\pi \, dx = -14\pi x$$
$$\int 3x^{-1} \, dx = 3 \int x^{-1} \, dx = 3 \ln|x|$$
$$\int -6x^{-4} \, dx = -6 \frac{x^{-4+1}}{-4+1} = -6 \frac{x^{-3}}{-3} = 2x^{-3} = \frac{2}{x^3}$$
**Step 3:** Combine:
$$\int\left(-14\pi + \frac{3}{x} - \frac{6}{x^4}\right)\,dx = -14\pi x + 3 \ln|x| + \frac{2}{x^3} + C$$
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3. **Problem:** Integrate $\int(7 - 9x^2)^2 \, dx$.
**Step 1:** Expand the square:
$$(7 - 9x^2)^2 = 49 - 2 \cdot 7 \cdot 9x^2 + (9x^2)^2 = 49 - 126x^2 + 81x^4$$
**Step 2:** Integrate term by term:
$$\int 49 \, dx = 49x$$
$$\int -126x^2 \, dx = -126 \frac{x^{3}}{3} = -42x^3$$
$$\int 81x^4 \, dx = 81 \frac{x^{5}}{5} = \frac{81}{5} x^5$$
**Step 3:** Combine:
$$\int(7 - 9x^2)^2 \, dx = 49x - 42x^3 + \frac{81}{5} x^5 + C$$
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4. **Problem:** Find $\int \frac{-15x^5 + x^3 + 10x^2}{x^3} \, dx$.
**Step 1:** Simplify the integrand by dividing each term by $x^3$:
$$\frac{-15x^5}{x^3} = -15x^{5-3} = -15x^2$$
$$\frac{x^3}{x^3} = 1$$
$$\frac{10x^2}{x^3} = 10x^{2-3} = 10x^{-1}$$
So integrand is $-15x^2 + 1 + 10x^{-1}$.
**Step 2:** Integrate term by term:
$$\int -15x^2 \, dx = -15 \frac{x^{3}}{3} = -5x^3$$
$$\int 1 \, dx = x$$
$$\int 10x^{-1} \, dx = 10 \ln|x|$$
**Step 3:** Combine:
$$\int \frac{-15x^5 + x^3 + 10x^2}{x^3} \, dx = -5x^3 + x + 10 \ln|x| + C$$
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5. **Problem:** Find $\int \left(\frac{3}{x^3} + \frac{2}{\sqrt[3]{x}}\right) \, dx$.
**Step 1:** Rewrite terms with powers:
$$\frac{3}{x^3} = 3x^{-3}, \quad \frac{2}{\sqrt[3]{x}} = 2x^{-\frac{1}{3}}$$
**Step 2:** Integrate each term:
$$\int 3x^{-3} \, dx = 3 \frac{x^{-3+1}}{-3+1} = 3 \frac{x^{-2}}{-2} = -\frac{3}{2} x^{-2} = -\frac{3}{2x^2}$$
$$\int 2x^{-\frac{1}{3}} \, dx = 2 \frac{x^{-\frac{1}{3}+1}}{-\frac{1}{3}+1} = 2 \frac{x^{\frac{2}{3}}}{\frac{2}{3}} = 2 \cdot \frac{3}{2} x^{\frac{2}{3}} = 3x^{\frac{2}{3}}$$
**Step 3:** Combine:
$$\int \left(\frac{3}{x^3} + \frac{2}{\sqrt[3]{x}}\right) \, dx = -\frac{3}{2x^2} + 3x^{\frac{2}{3}} + C$$
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6. **Problem:** Find $\int \sqrt{x} \left(x + \frac{2}{x}\right) \, dx$.
**Step 1:** Rewrite $\sqrt{x} = x^{\frac{1}{2}}$ and distribute:
$$x^{\frac{1}{2}} \cdot x = x^{\frac{1}{2} + 1} = x^{\frac{3}{2}}$$
$$x^{\frac{1}{2}} \cdot \frac{2}{x} = 2x^{\frac{1}{2} - 1} = 2x^{-\frac{1}{2}}$$
So integrand is $x^{\frac{3}{2}} + 2x^{-\frac{1}{2}}$.
**Step 2:** Integrate term by term:
$$\int x^{\frac{3}{2}} \, dx = \frac{x^{\frac{3}{2} + 1}}{\frac{3}{2} + 1} = \frac{x^{\frac{5}{2}}}{\frac{5}{2}} = \frac{2}{5} x^{\frac{5}{2}}$$
$$\int 2x^{-\frac{1}{2}} \, dx = 2 \frac{x^{-\frac{1}{2} + 1}}{-\frac{1}{2} + 1} = 2 \frac{x^{\frac{1}{2}}}{\frac{1}{2}} = 2 \cdot 2 x^{\frac{1}{2}} = 4 \sqrt{x}$$
**Step 3:** Combine:
$$\int \sqrt{x} \left(x + \frac{2}{x}\right) \, dx = \frac{2}{5} x^{\frac{5}{2}} + 4 \sqrt{x} + C$$