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قم بحل المعادلة مع شرح طريقة الحل

Let's solve each of these integrals step by step with detailed explanations.

### 1. \(\int \frac{dx}{x^2}\)

Rewrite the integrand:
\[
\int x^{-2} \, dx
\]

Use the power rule for integration:
\[
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad \text{for } n \neq -1
\]

Here, \(n = -2\):
\[
\int x^{-2} \, dx = \frac{x^{-2+1}}{-2+1} + C = \frac{x^{-1}}{-1} + C = -\frac{1}{x} + C
\]

So,
\[
\int \frac{dx}{x^2} = -\frac{1}{x} + C
\]

### 2. \(\int x^{\sqrt{2}} \, dx\)

Use the power rule for integration:
\[
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad \text{for } n \neq -1
\]

Here, \(n = \sqrt{2}\):
\[
\int x^{\sqrt{2}} \, dx = \frac{x^{\sqrt{2}+1}}{\sqrt{2}+1} + C
\]

So,
\[
\int x^{\sqrt{2}} \, dx = \frac{x^{\sqrt{2}+1}}{\sqrt{2}+1} + C
\]

### 3. \(\int \frac{1}{x^\pi} \, dx\)

Rewrite the integrand:
\[
\int x^{-\pi} \, dx
\]

Use the power rule for integration:
\[
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad \text{for } n \neq -1
\]

Here, \(n = -\pi\):
\[
\int x^{-\pi} \, dx = \frac{x^{-\pi+1}}{-\pi+1} + C
\]

So,
\[
\int \frac{1}{x^\pi} \, dx = \frac{x^{1-\pi}}{1-\pi} + C
\]

### 4. \(\int x\sqrt{x} \, dx\)

Rewrite \(\sqrt{x}\) as \(x^{1/2}\):
\[
\int x \cdot x^{1/2} \, dx = \int x^{1 + 1/2} \, dx = \int x^{3/2} \, dx
\]

Use the power rule for integration:
\[
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad \text{for } n \neq -1
\]

Here, \(n = \frac{3}{2}\):
\[
\int x^{3/2} \, dx = \frac{x^{3/2 + 1}}{3/2 + 1} + C = \frac{x^{5/2}}{5/2} + C = \frac{2}{5} x^{5/2} + C
\]

So,
\[
\int x\sqrt{x} \, dx = \frac{2}{5} x^{5/2} + C
\]

If you have any questions or need further details, feel free to ask.

Here are 5 related questions you can ask:
1. How do you solve \(\int \frac{1}{x} \, dx\)?
2. Can you explain the substitution method in integration?
3. What is the integration by parts formula?
4. How do you integrate trigonometric functions?
5. Can you solve a definite integral, like \(\int_0^1 x^2 \, dx\)?

**Tip:** Always remember to add the constant of integration \(C\) when solving indefinite integrals.

Learn how to solve integration problems step-by-step with detailed explanations. This page covers various types of integrals including exponential functions and demonstrates the power rule for integration. Suitable for advanced high school and college-level students.

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