3.2 Q-1

Question Statement

Evaluate the following indefinite integrals:

i. $\int\left(3 x^{2}-2 x+1\right) dx$

ii. $\int\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right) dx, x>0$

iii. $\int x(\sqrt{x+1}) dx, x>0$

iv. $\int(2 x+3)^{\frac{1}{2}} dx$

v. $\int(\sqrt{x}+1)^{2} dx$

vi. $\int\left(\sqrt{x}-\frac{1}{\sqrt{x}}\right)^{2} dx$

vii. $\int \frac{3 x+2}{\sqrt{x}} dx, x>0$

viii. $\int \frac{\sqrt{y}(y+1)}{y} dy, y>0$

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Background and Explanation

To solve these indefinite integrals, we need to recall some basic integration rules and strategies:

1. Power Rule for Integration: For any function of the form $x^n$, the integral is given by:

$$\int x^n dx = \frac{x^{n+1}}{n+1} + C, \quad \text{(for } n \neq -1\text{)}$$

2. Sum of Integrals: The integral of a sum of functions is the sum of their integrals:

$$\int \left( f(x) + g(x) \right) dx = \int f(x) dx + \int g(x) dx$$

3. Special Integrals:
   • For integrals involving square roots, we use the identity $\sqrt{x} = x^{1/2}$.
   • Use of substitution where necessary, such as when expressions can be simplified by expanding or factoring terms.

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Solution

i. $\int\left(3 x^{2}-2 x+1\right) dx$

Step-by-step:

1. Split the integral:

$$\int\left(3 x^{2}-2 x+1\right) dx = \int 3 x^2 dx - \int 2x dx + \int 1 dx$$

2. Integrate each term:

   • $\int 3 x^2 dx = 3 \cdot \frac{x^3}{3} = x^3$
   • $\int 2x dx = 2 \cdot \frac{x^2}{2} = x^2$
   • $\int 1 dx = x$

3. Combine results:

$$\boxed{x^3 - x^2 + x + C}$$

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ii. $\int\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right) dx, x>0$

Step-by-step:

1. Write as powers of $x$:

$$\int\left(\sqrt{x} + \frac{1}{\sqrt{x}}\right) dx = \int x^{1/2} dx + \int x^{-1/2} dx$$

2. Apply the power rule:

   • $\int x^{1/2} dx = \frac{x^{3/2}}{3/2} = \frac{2}{3} x^{3/2}$
   • $\int x^{-1/2} dx = \frac{x^{1/2}}{1/2} = 2x^{1/2}$

3. Combine results:

$$\boxed{\frac{2}{3} x^{3/2} + 2x^{1/2} + C}$$

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iii. $\int x(\sqrt{x+1}) dx, x>0$

Step-by-step:

1. Expand the integrand:

$$\int x(\sqrt{x+1}) dx = \int \left(x(x+1)^{1/2}\right) dx = \int x^{3/2} dx + \int x dx$$

2. Integrate each term:

   • $\int x^{3/2} dx = \frac{2}{5} x^{5/2}$
   • $\int x dx = \frac{x^2}{2}$

3. Combine results:

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

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iv. $\int(2 x+3)^{\frac{1}{2}} dx$

Step-by-step:

1. Apply substitution $u = 2x + 3$, hence $du = 2 dx$:

$$\frac{1}{2} \int (u)^{1/2} du$$

2. Integrate:

   • $\int u^{1/2} du = \frac{2}{3} u^{3/2}$

3. Substitute back $u = 2x + 3$:

$$\boxed{\frac{1}{3} (2x+3)^{3/2} + C}$$

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v. $\int(\sqrt{x}+1)^{2} dx$

Step-by-step:

1. Expand the integrand:

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

2. Integrate each term:

   • $\int x dx = \frac{x^2}{2}$
   • $\int 2x^{1/2} dx = \frac{4}{3} x^{3/2}$
   • $\int 1 dx = x$

3. Combine results:

$$\boxed{\frac{x^2}{2} + \frac{4}{3} x^{3/2} + x + C}$$

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vi. $\int\left(\sqrt{x}-\frac{1}{\sqrt{x}}\right)^{2} dx$

Step-by-step:

1. Expand the integrand:

$$\left(\sqrt{x}-\frac{1}{\sqrt{x}}\right)^2 = x - 2 + \frac{1}{x}$$

2. Integrate each term:

   • $\int x dx = \frac{x^2}{2}$
   • $\int -2 dx = -2x$
   • $\int \frac{1}{x} dx = \ln |x|$

3. Combine results:

$$\boxed{\frac{x^2}{2} - 2x + \ln |x| + C}$$

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vii. $\int \frac{3 x+2}{\sqrt{x}} dx, x>0$

Step-by-step:

1. Simplify:

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

2. Integrate each term:

   • $\int 3 x^{1/2} dx = 2x^{3/2}$
   • $\int 2 x^{-1/2} dx = 4x^{1/2}$

3. Combine results:

$$\boxed{2x^{3/2} + 4x^{1/2} + C}$$

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viii. $\int \frac{\sqrt{y}(y+1)}{y} dy, y>0$

Step-by-step:

1. Simplify:

$$\int \frac{\sqrt{y}(y+1)}{y} dy = \int \left(y^{1/2} + y^{-1/2}\right) dy$$

2. Integrate each term:

   • $\int y^{1/2} dy = \frac{2}{3} y^{3/2}$
   • $\int y^{-1/2} dy = 2y^{1/2}$

3. Combine results:

$$\boxed{\frac{2}{3} y^{3/2} + 2y^{1/2} + C}$$

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Key Formulas or Methods Used

• Power Rule for Integration: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$
• Sum of Integrals: $\int (f(x) + g(x)) dx = \int f(x) dx + \int g(x) dx$

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Summary of Steps

1. Split the integrand into manageable parts.
2. Apply the power rule for each term.
3. Combine the results and include the constant of integration, $C$.

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