Definite Integral - Miscellaneous Problems ( Related To Complex Function) , Solving Them With Certain Properties

In this lecture, we will be solving miscellaneous problems related to complex functions using definite integral and certain properties.
- Example 1: Find the value of $\int_{0}^{\infty} e^{-ax}\cos(bx) , dx$, with $a>0$ and $b^2>a$.
  - Solution:
    - Let’s consider the function $I(a) = \int_{0}^{\infty} e^{-ax}\cos(bx) , dx$.
    - Differentiating both sides of the equation with respect to $a$, we get:
    - $\frac{d}{da} I(a) = -\int_{0}^{\infty} xe^{-ax}\cos(bx) , dx$
    - Now, differentiating the above equation again, we have:
    - $\frac{d^2}{da^2} I(a) = \int_{0}^{\infty} x^2e^{-ax}\cos(bx) , dx$
    - Substituting $a = \frac{1}{b^2}$ and integrating by parts, we can obtain the final value.
- Example 2: Evaluate $\int_{0}^{1} \frac{\ln(1+x)}{1+x^2} , dx$.
  - Solution:
    - We can first make the substitution $x = \tan(t)$.
    - This will transform the integral into $\int_{0}^{\frac{\pi}{4}} \ln(1+\tan(t)) , dt$.
    - Then, we can use the properties of definite integrals and certain trigonometric identities to simplify the expression.
- Example 3: Calculate $\int_{0}^{\pi} \frac{x}{1+\sin(x)} , dx$.
  - Solution:
    - Using the substitution $u = \pi - x$, we can convert the integral into $\int_{0}^{\pi} \frac{\pi - u}{1 + \sin(u)} , du$.
    - Adding the two integrals together, we get $\int_{0}^{\pi} \pi , du$.
    - Simplifying further, the integral evaluates to $\pi^2$.
- Example 4: Find $\int_{0}^{1} \frac{\ln(x)}{\sqrt{1-x^2}} , dx$.
  - Solution:
    - Using the substitution $x = \sin(t)$, we can rewrite the integral as $\int_{0}^{\frac{\pi}{2}} \frac{\ln(\sin(t))}{\cos(t)} , dt$.
    - Applying certain trigonometric identities and using properties of definite integrals, we will obtain the final result.
- Example 5: Determine $\int_{0}^{1} \frac{x^3}{(1-x^2)^4} , dx$.
  - Solution:
    - By using partial fraction decomposition, we can break the integrand into simpler fractions.
    - This will allow us to solve each fraction separately and obtain the final result. 11. Definite Integral - Miscellaneous Problems (Related to Complex Function) - Examples
Example 1: Find the value of $\int_{0}^{\infty} e^{-ax}\cos(bx) , dx$, with $a>0$ and $b^2>a$.
- Solution:
  - Let’s consider the function $I(a) = \int_{0}^{\infty} e^{-ax}\cos(bx) , dx$.
  - Differentiating both sides of the equation with respect to $a$, we get: $\frac{d}{da} I(a) = -\int_{0}^{\infty} xe^{-ax}\cos(bx) , dx$.
  - Now, differentiating the above equation again, we have: $\frac{d^2}{da^2} I(a) = \int_{0}^{\infty} x^2e^{-ax}\cos(bx) , dx$.
  - Substituting $a = \frac{1}{b^2}$ and integrating by parts, we can obtain the final value.
Example 2: Evaluate $\int_{0}^{1} \frac{\ln(1+x)}{1+x^2} , dx$.
- Solution:
  - We can first make the substitution $x = \tan(t)$.
  - This will transform the integral into $\int_{0}^{\frac{\pi}{4}} \ln(1+\tan(t)) , dt$.
  - Then, we can use the properties of definite integrals and certain trigonometric identities to simplify the expression.
Example 3: Calculate $\int_{0}^{\pi} \frac{x}{1+\sin(x)} , dx$.
- Solution:
  - Using the substitution $u = \pi - x$, we can convert the integral into $\int_{0}^{\pi} \frac{\pi - u}{1 + \sin(u)} , du$.
  - Adding the two integrals together, we get $\int_{0}^{\pi} \pi , du$.
  - Simplifying further, the integral evaluates to $\pi^2$.
Example 4: Find $\int_{0}^{1} \frac{\ln(x)}{\sqrt{1-x^2}} , dx$.
- Solution:
  - Using the substitution $x = \sin(t)$, we can rewrite the integral as $\int_{0}^{\frac{\pi}{2}} \frac{\ln(\sin(t))}{\cos(t)} , dt$.
  - Applying certain trigonometric identities and using properties of definite integrals, we will obtain the final result.
Example 5: Determine $\int_{0}^{1} \frac{x^3}{(1-x^2)^4} , dx$.
- Solution:
  - By using partial fraction decomposition, we can break the integrand into simpler fractions.
  - This will allow us to solve each fraction separately and obtain the final result.

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Definite Integral - Miscellaneous Problems (Related to Complex Function)

Examples
1. Find the value of $\int_{0}^{\infty} e^{-ax}\cos(bx) , dx$, with $a>0$ and $b^2>a$.
  - Solution:
    - Let’s consider the function $I(a) = \int_{0}^{\infty} e^{-ax}\cos(bx) , dx$.
    - Differentiating both sides of the equation with respect to $a$, we get:
    - $\frac{d}{da} I(a) = -\int_{0}^{\infty} xe^{-ax}\cos(bx) , dx$
    - Now, differentiating the above equation again, we have:
    - $\frac{d^2}{da^2} I(a) = \int_{0}^{\infty} x^2e^{-ax}\cos(bx) , dx$
    - Substituting $a = \frac{1}{b^2}$ and integrating by parts, we can obtain the final value.
2. Evaluate $\int_{0}^{1} \frac{\ln(1+x)}{1+x^2} , dx$.
  - Solution:
    - We can first make the substitution $x = \tan(t)$.
    - This will transform the integral into $\int_{0}^{\frac{\pi}{4}} \ln(1+\tan(t)) , dt$.
    - Then, we can use the properties of definite integrals and certain trigonometric identities to simplify the expression.

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Definite Integral - Miscellaneous Problems (Related to Complex Function)

Examples 3. Calculate $\int_{0}^{\pi} \frac{x}{1+\sin(x)} , dx$.
```
  - Solution:
      - Using the substitution $u = \pi - x$, we can convert the integral into $\int_{0}^{\pi} \frac{\pi - u}{1 + \sin(u)} \, du$.
      - Adding the two integrals together, we get $\int_{0}^{\pi} \pi \, du$.
      - Simplifying further, the integral evaluates to $\pi^2$.
```
1. Find $\int_{0}^{1} \frac{\ln(x)}{\sqrt{1-x^2}} , dx$.
  - Solution:
    - Using the substitution $x = \sin(t)$, we can rewrite the integral as $\int_{0}^{\frac{\pi}{2}} \frac{\ln(\sin(t))}{\cos(t)} , dt$.
    - Applying certain trigonometric identities and using properties of definite integrals, we will obtain the final result.
2. Determine $\int_{0}^{1} \frac{x^3}{(1-x^2)^4} , dx$.
  - Solution:
    - By using partial fraction decomposition, we can break the integrand into simpler fractions.
    - This will allow us to solve each fraction separately and obtain the final result.

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Definite Integral - Miscellaneous Problems (Related to Complex Function)

Example 1 Summary:
- $\int_{0}^{\infty} e^{-ax}\cos(bx) , dx$
- $a>0$ and $b^2>a$
- Differentiate twice to obtain the final value
Example 2 Summary:
- $\int_{0}^{1} \frac{\ln(1+x)}{1+x^2} , dx$
- Substitute $x = \tan(t)$ and simplify using properties of definite integrals and trigonometric identities

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Definite Integral - Miscellaneous Problems (Related to Complex Function)

Example 3 Summary:
- $\int_{0}^{\pi} \frac{x}{1+\sin(x)} , dx$
- Use the substitution $u = \pi - x$ and simplify to obtain the final result
Example 4 Summary:
- $\int_{0}^{1} \frac{\ln(x)}{\sqrt{1-x^2}} , dx$
- Substitute $x = \sin(t)$ and apply trigonometric identities and properties of definite integrals
Example 5 Summary:
- $\int_{0}^{1} \frac{x^3}{(1-x^2)^4} , dx$
- Break the integrand into partial fractions and solve separately to obtain the final result

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Definite Integral - Miscellaneous Problems (Related to Complex Function)

Recap of the Examples
- Example 1:
  - Value of $\int_{0}^{\infty} e^{-ax}\cos(bx) , dx$ for $a>0$ and $b^2>a$
- Example 2:
  - Evaluation of $\int_{0}^{1} \frac{\ln(1+x)}{1+x^2} , dx$
- Example 3:
  - Calculation of $\int_{0}^{\pi} \frac{x}{1+\sin(x)} , dx$

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Definite Integral - Miscellaneous Problems (Related to Complex Function)

Recap of the Examples
- Example 4:
  - Finding $\int_{0}^{1} \frac{\ln(x)}{\sqrt{1-x^2}} , dx$
- Example 5:
  - Determining $\int_{0}^{1} \frac{x^3}{(1-x^2)^4} , dx$

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Definite Integral - Miscellaneous Problems (Related to Complex Function)

Key Takeaways
- Definite integrals can be used to solve a wide range of problems related to complex functions.
- Properties of definite integrals, along with certain substitutions and trigonometric identities, can simplify the calculations.
- Differentiating and integrating twice can help solve specific types of problems.

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Definite Integral - Miscellaneous Problems (Related to Complex Function)

Key Takeaways
- Substitution technique, such as $x = \tan(t)$ or $x = \sin(t)$, can often simplify the integration process.
- Breaking the integrand into partial fractions allows us to solve each fraction separately and obtain the final result.
- It is important to carefully analyze the conditions and properties of the given integral to determine the appropriate approach.

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Definite Integral - Miscellaneous Problems (Related to Complex Function)

Practice Problems
1. Find the value of $\int_{0}^{\infty} e^{-2x}\sin(3x) , dx$.
2. Evaluate $\int_{0}^{1} \frac{x^2}{\sqrt{1-x^2}} , dx$.
3. Calculate $\int_{0}^{2\pi} \frac{\sin^2(x)}{1+\cos^2(x)} , dx$.
4. Find $\int_{0}^{1} \frac{1}{x^2 + 1} , dx$.
5. Determine $\int_{0}^{\frac{\pi}{2}} \frac{\sin(2x)}{1+\cos^2(x)} , dx$.

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Definite Integral - Miscellaneous Problems (Related to Complex Function)

Stay curious and keep practicing!
Practice makes perfect, so make sure to attempt more problems and explore different approaches.
If you have any questions or need further clarification, feel free to ask.
Good luck with your studies and remember, maths is all about continuous learning and exploration!