3.3 Q-6

Question Statement

Evaluate the following integral:

$$\int \frac{x+b}{\left(x^{2}+2 b x + C\right)^{\frac{1}{2}}} , dx$$

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Background and Explanation

This integral involves a rational expression with a square root in the denominator. The solution uses substitution and algebraic manipulation to simplify the expression, eventually leading to a standard form that can be integrated. The general approach here involves recognizing that the expression inside the square root can be integrated using known techniques.

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Solution

1. Rearrange the integral:
   Start by rewriting the given integral:

$$\int \frac{x+b}{\left(x^{2}+2 b x + C\right)^{\frac{1}{2}}} , dx$$

You can express this as:

$$\int \left(x^{2}+2bx + C\right)^{\frac{1}{2}}(x+b) , dx$$

2. Use substitution:
   Notice that we can simplify the integrand by recognizing the form of the derivative of $x^2 + 2bx + C$. Let’s multiply and divide the integrand by 2:

$$= \frac{1}{2} \int \left(x^2 + 2bx + C\right)^{\frac{1}{2}} \left(2x + 2b\right) , dx$$

3. Simplify the integration:
   The expression now fits the form of a standard integral. Integrating this, we get:

$$= \frac{1}{2} \frac{\left(x^2 + 2bx + C\right)^{\frac{1}{2}+1}}{\frac{1}{2}+1} + C_1$$

4. Simplify further:
   Simplify the exponent and denominator:

$$= \frac{1}{2} \frac{\left(x^2 + 2bx + C\right)^{\frac{3}{2}}}{\frac{3}{2}} + C_1$$

This simplifies to:

$$= \frac{2}{3} \left(x^2 + 2bx + C\right)^{\frac{3}{2}} + C_1$$

Which is the final solution after performing the integration.

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Key Formulas or Methods Used

• Standard Integral Form:
  This solution applies the standard method of integrating expressions of the form $(x^2 + 2bx + C)^{\frac{1}{2}}$.

• Substitution:
  Recognizing the structure of the integral and applying algebraic manipulation by introducing a factor of 2 to make the integration easier.

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Summary of Steps

1. Rewrite the integral to a more manageable form.
2. Use substitution and factor out constants.
3. Apply the standard integral formula.
4. Simplify the result to obtain the final solution.