03_Ex 3.3

Exercise Questions

Questions	Question Links
Q1. $\int \frac{-2 \mathrm{x}}{\sqrt{4-\mathrm{x}^{2}}} \mathrm{dx}$	3.3 Q-1
Q2. $\int \frac{\mathrm{dx}}{\mathrm{x}^{2}+4 x+13} \mathrm{dx}$	3.3 Q-2
Q3. $\int \frac{\mathrm{x}^{2}}{4+\mathrm{x}^{2}} \mathrm{dx}$	3.3 Q-3
Q4. $\int \frac{1}{\mathrm{x} \ln \mathrm{x}} \mathrm{dx}$	3.3 Q-4
Q5. $\int \frac{e^{\mathrm{x}}}{e^{x}+3} \mathrm{dx}$	3.3 Q-5
Q6. $\int \frac{x+b}{\left(x^{2}+2 b x+C\right)^{\frac{1}{2}}} d x$	3.3 Q-6
Q7. $\frac{\operatorname{Sec}^{2} x}{\sqrt{\tan x}} d x$	3.3 Q-7
Q8. Show that…	3.3 Q-8
Q9. $\int \frac{d x}{\left(1+x^{2}\right)^{\frac{3}{2}}}$	3.3 Q-9
Q10. $\int \frac{1}{\left(1+x^{2}\right) \tan 1 x} d x$	3.3 Q-10
Q11. $\int \sqrt{\frac{1+\mathrm{x}}{1-\mathrm{x}}} \mathrm{dx}$	3.3 Q-11
Q12. $\int \frac{\operatorname{Sin} \theta \mathrm{d} \theta}{1+\operatorname{Cos}^{2} \theta}$	3.3 Q-12
Q13. $\int \frac{\mathrm{ax}}{\sqrt{\mathrm{a}^{2}-\mathrm{x}^{4}}} \mathrm{dx}$	3.3 Q-13
Q14. $\int \frac{d x}{\sqrt{7-6 x-x^{2}}}$	3.3 Q-14
Q15. $\int \frac{\cos \mathrm{x}}{\sin \mathrm{xln} \sin \mathrm{x}} \mathrm{dx}$	3.3 Q-15
Q16. $\int \operatorname{Cos} x\left(\frac{\ln \sin x}{\operatorname{Sin} x}\right) d x$	3.3 Q-16
Q17. $\int \frac{\mathrm{xdx}}{4+2 \mathrm{x}+\mathrm{x}^{2}}$	3.3 Q-17
Q18. $\int \frac{\mathrm{xdx}}{\mathrm{x}^{4}+2 \mathrm{x}^{2}+5}$	3.3 Q-18
Q19. $\int\left[\operatorname{Cos}\left(\sqrt{\mathrm{x}}-\frac{x}{2}\right)\right] \times\left[\frac{1}{\sqrt{\mathrm{x}}}-1\right] d x$	3.3 Q-19
Q20. $\int \frac{x+2}{\sqrt{\mathrm{x}+3}} \mathrm{dx}$	3.3 Q-20
Q21. $\int \frac{\sqrt{3}}{\operatorname{Sin} \mathrm{x}+\operatorname{Cos} \mathrm{x}} \mathrm{dx}$	3.3 Q-21
Q22. $\int \frac{\mathrm{dx}}{\frac{1}{2} \sin \mathrm{x}+\frac{\sqrt{3}}{2} \operatorname{Cos} \mathrm{x}}$	3.3 Q-22

Overview

This exercise focuses on various integration techniques, including substitution, trigonometric identities, and standard integral formulas. These techniques are crucial for solving more complex integrals and are widely used in fields such as physics, engineering, and economics to model various real-world situations.

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Key Concepts

• Integral: A fundamental concept in calculus that represents the area under a curve or the accumulation of quantities.
• Definite Integral: $\int_a^b f(x) , dx$, representing the area under the curve between the limits $a$ and $b$.
• Indefinite Integral: $\int f(x) , dx = F(x) + C$, where $C$ is the constant of integration.
• Antiderivative: A function $F(x)$ such that $F'(x) = f(x)$.
• Substitution Method: A technique used to simplify integrals by substituting a part of the integrand with a new variable.

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Important Formulas

• Standard Integrals:

  • $\int \frac{1}{a^2 + x^2} , dx = \frac{1}{a} \tan^{-1} \left( \frac{x}{a} \right) + C$
  • $\int \frac{1}{\sqrt{a^2 - x^2}} , dx = \sin^{-1} \left( \frac{x}{a} \right) + C$
  • $\int \sqrt{a^2 - x^2} , dx = \frac{x}{2} \sqrt{a^2 - x^2} + \frac{a^2}{2} \sin^{-1} \left( \frac{x}{a} \right) + C$
  • $\int e^{ax} , dx = \frac{1}{a} e^{ax} + C$

• Trigonometric Substitutions:

  1. For $\sqrt{a^2 - x^2}$ use the substitution $x = a \sin(\theta)$.

  2. For $\sqrt{x^2 - a^2}$ use the substitution $x = a \sec(\theta)$.

  3. For $\sqrt{a^2 + x^2}$ use the substitution $x = a \tan(\theta)$.

• Integration by Parts: $\int u , dv = uv - \int v , du$

• Partial Fraction Decomposition: Used for breaking down complex rational expressions into simpler terms for easier integration.

• Trigonometric Identities:

  • $\sin^2(x) + \cos^2(x) = 1$
  • $1 + \tan^2(x) = \sec^2(x)$
  • $1 + \cot^2(x) = \csc^2(x)$

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Tips and Tricks

• Recognizing Integral Patterns: Spot standard integral forms such as $\int \frac{1}{a^2 + x^2}$ and use the corresponding formulas for quicker solutions.

• Using Substitutions: When encountering integrals with square roots or rational functions, use substitution methods to simplify the expression. For instance, when dealing with $\sqrt{a^2 - x^2}$, use the substitution $x = a \sin(\theta)$ to convert the integral into a simpler form.

• Trigonometric Substitutions: If the integral involves a square root of a quadratic expression, look for opportunities to apply trigonometric identities to simplify the integral.

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Summary

This exercise covers several critical integration techniques, such as substitution, integration by parts, and partial fractions Although a deeper dive on Integration by the method of partial fractions can be found on 05_Ex 3.5.

By mastering these techniques and formulas, students can solve a wide range of integrals that arise in various applications, especially in physics and engineering.

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Reference

By Sir Shahzad Sair:

By Great Science Academy:

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