indiqués: $$ (6 $$ pts) 10) $$ F(x)=\int 2 x \cdot\left(1+x^{2}\right)^{2} d x ; F(0)=0 $$ 20) $$ F(x)=\int\left(\frac{3}{\sqrt{x}}-\frac{1}{x^{2}}+\sqrt{x}+3\right) d x ; F(1)=0 $$ 3) $$ F(x)=\int\left(\frac{1}{x \cdot \sqrt{x}}+x \cdot \sqrt{x}\right) d x \quad ; F(1)=1 $$

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Let's solve each integral step by step.

### 10) $$ F(x)=\int 2 x \cdot\left(1+x^{2}\right)^{2} d x ; F(0)=0 $$

1. **Expand the integrand**:
   $$
   2x(1+x^2)^2 = 2x(1 + 2x^2 + x^4) = 2x + 4x^3 + 2x^5
   $$

2. **Integrate term by term**:
   $$
   F(x) = \int (2x + 4x^3 + 2x^5) \, dx = x^2 + x^4 + \frac{2}{6}x^6 + C = x^2 + x^4 + \frac{1}{3}x^6 + C
   $$

3. **Apply the condition $$ F(0) = 0 $$**:
   $$
   F(0) = 0^2 + 0^4 + \frac{1}{3}(0)^6 + C = C \implies C = 0
   $$

Thus, 
$$
F(x) = x^2 + x^4 + \frac{1}{3}x^6
$$

### 20) $$ F(x)=\int\left(\frac{3}{\sqrt{x}}-\frac{1}{x^{2}}+\sqrt{x}+3\right) d x ; F(1)=0 $$

1. **Integrate term by term**:
   $$
   F(x) = \int \left(3x^{-1/2} - x^{-2} + x^{1/2} + 3\right) \, dx
   $$
   - The integrals are:
     - $$ \int 3x^{-1/2} \, dx = 6\sqrt{x} $$
     - $$ \int -x^{-2} \, dx = \frac{1}{x} $$
     - $$ \int x^{1/2} \, dx = \frac{2}{3}x^{3/2} $$
     - $$ \int 3 \, dx = 3x $$

2. **Combine the results**:
   $$
   F(x) = 6\sqrt{x} - \frac{1}{x} + \frac{2}{3}x^{3/2} + 3x + C
   $$

3. **Apply the condition $$ F(1) = 0 $$**:
   $$
   F(1) = 6(1) - 1 + \frac{2}{3}(1) + 3(1) + C = 6 - 1 + \frac{2}{3} + 3 + C = 8 + \frac{2}{3} + C
   $$
   $$
   8 + \frac{2}{3} + C = 0 \implies C = -8 - \frac{2}{3} = -\frac{24}{3} - \frac{2}{3} = -\frac{26}{3}
   $$

Thus, 
$$
F(x) = 6\sqrt{x} - \frac{1}{x} + \frac{2}{3}x^{3/2} + 3x - \frac{26}{3}
$$

### 30) $$ F(x)=\int\left(\frac{1}{x \cdot \sqrt{x}}+x \cdot \sqrt{x}\right) d x \quad ; F(1)=1 $$

1. **Simplify the integrand**:
   $$
   \frac{1}{x \cdot \sqrt{x}} = x^{-3/2}, \quad x \cdot \sqrt{x} = x^{3/2}
   $$

2. **Integrate term by term**:
   $$
   F(x) = \int \left(x^{-3/2} + x^{3/2}\right) \, dx
   $$
   - The integrals are:
     - $$ \int x^{-3/2} \, dx = -2x^{-1/2} $$
     - $$ \int x^{3/2} \, dx = \frac{2}{5}x^{5/2} $$

3. **Combine the results**:
   $$
   F(x) = -2x^{-1/2} + \frac{2}{5}x^{5/2} + C
   $$

4. **Apply the condition $$ F(1) = 1 $$**:
   $$
   F(1) = -2(1) + \frac{2}{5}(1) + C = -2 + \frac{2}{5} + C = -\frac{10}{5} + \frac{2}{5} + C = -\frac{8}{5} + C
   $$
   $$
   -\frac{8}{5} + C = 1 \implies C = 1 + \frac{8}{5} = \frac{5}{5} + \frac{8}{5} = \frac{13}{5}
   $$

Thus, 
$$
F(x) = -2x^{-1/2} + \frac{2}{5}x^{5/2} + \frac{13}{5}
$$

### Summary of Results:
1. $$ F(x) = x^2 + x^4 + \frac{1}{3}x^6 $$
2. $$ F(x) = 6\sqrt{x} - \frac{1}{x} + \frac{2}{3}x^{3/2} + 3x - \frac{26}{3} $$
3. $$ F(x) = -2x^{-1/2} + \frac{2}{5}x^{5/2} + \frac{13}{5} $$
1. $$ F(x) = x^2 + x^4 + \frac{1}{3}x^6 $$ 2. $$ F(x) = 6\sqrt{x} - \frac{1}{x} + \frac{2}{3}x^{3/2} + 3x - \frac{26}{3} $$ 3. $$ F(x) = -2x^{-1/2} + \frac{2}{5}x^{5/2} + \frac{13}{5} $$

indiqués: pts)
10)
20)
3)

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10)

20)

30)

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indiqués: pts) 10) 20) 3)