06_Ex 3.6

Exercise Questions

Questions	Question Links
Q1. $\int_{1}^{2}\left(x^{2}+1\right) d x$	3.6 Q-1
Q2. $\int_{1}^{1}\left(x^{1 / 3}+1\right) d x$	3.6 Q-2
Q3. $\int_{-2}^{0} \frac{1}{(2 x-1)^{2}} d x$	3.6 Q-3
Q4. $\int_{-6}^{2} \sqrt{3-x} \mathrm{dx}$	3.6 Q-4
Q5. $\int_{1}^{\sqrt{5}} \sqrt{(2 t-1)^{3}} d x$	3.6 Q-5
Q6. $\int_{2}^{\sqrt{5}} x \sqrt{\mathrm{x}^{2}-1} \mathrm{dx}$	3.6 Q-6
Q7. $\int_{1}^{2} \frac{x}{x^{2}+2} d x$	3.6 Q-7
Q8. $\int_{2}^{3}\left(\mathrm{x}-\frac{1}{\mathrm{x}}\right)^{2} \mathrm{dx}$	3.6 Q-8
Q9. $\int_{-1}^{1}\left(x+\frac{1}{2}\right) \sqrt{x^{2}+x+1} \mathrm{dx}$	3.6 Q-9
Q10. $\int_{0}^{3} \frac{d x}{x^{2}+9}$	3.6 Q-10
Q11. $\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \operatorname{Cos} t \mathrm{dt}$	3.6 Q-11
Q12. $\int_{1}^{2}\left(\mathrm{x}+\frac{1}{\mathrm{x}}\right)^{\frac{1}{2}}\left(1-\frac{1}{\mathrm{x}^{2}}\right) \quad \mathrm{dx}$	3.6 Q-12
Q13. $\int_{1}^{2} \ln x d x$	3.6 Q-13
Q14. $\int_{0}^{2}\left(e^{x / 2}-e^{-x / 2}\right) d x$	3.6 Q-14
Q15. $\int_{0}^{\frac{\pi}{4}} \frac{\operatorname{Cos} \theta+\operatorname{Sin} \theta}{\operatorname{Cos} 2 \theta} d \theta$	3.6 Q-15
Q16. $\int_{0}^{\frac{\pi}{6}} \operatorname{Cos}^{3} \theta \mathrm{~d} \theta$	3.6 Q-16
Q17. $\int_{\frac{\pi}{6}}^{\frac{\pi}{4}} \operatorname{Cos}^{2} \theta \operatorname{Cot}^{2} \theta d \theta$	3.6 Q-17
Q18. $\int_{0}^{\frac{\pi}{4}} \operatorname{Cos}^{4} t d t$	3.6 Q-18
Q19. $\int_{0}^{\frac{\pi}{3}} \operatorname{Cos}^{2} \operatorname{Sin} \theta \mathrm{~d} \theta$	3.6 Q-19
Q20. $\int_{0}^{\frac{\pi}{4}}\left(1+\operatorname{Cos}^{2} \theta\right) \tan ^{2} \theta d \theta$	3.6 Q-20
Q21. $\int_{0}^{\frac{\pi}{4}} \frac{\operatorname{Sec} \theta}{\operatorname{Sin} \theta+\operatorname{Cos} \theta} \mathrm{d} \theta$	3.6 Q-21
Q22. $\int_-1^5	x-3
Q23. $\int_{1 / 8}^{1} \frac{\left(x^{1 / 3}+2\right)^{2}}{x^{2 / 3}} \mathrm{dx}$	3.6 Q-23
Q24. $\int_{1}^{3} \frac{\mathrm{x}^{2}-2}{\mathrm{x}+1} \mathrm{dx}$	3.6 Q-24
Q25. $\int_{2}^{3} \frac{3 x^{2}-2 x+1}{(x-1)\left(x^{2}+1\right)} d x$	3.6 Q-25
Q26. $\int_0^{\frac{\pi}{4}} \frac{\sin x-1}{\cos ^2 x} d x$	3.6 Q-26
Q27. $\int_{0}^{\pi / 4} \frac{1}{1+\operatorname{Sin} \mathrm{x}} \mathrm{dx}$	3.6 Q-27
Q28. $\int_{0}^{1} \frac{3 x}{\sqrt{4-3 x}} d x$	3.6 Q-28
Q29. $\int_{\pi / 6}^{\pi / 2} \frac{\operatorname{Cos} \mathrm{x}}{\operatorname{Sin} \mathrm{x}(2+\operatorname{Sin} \mathrm{x})} \mathrm{dx}$	3.6 Q-29
Q30. $\int_{0}^{2} \frac{\operatorname{Sin} x}{(1+\operatorname{Cos} x)(2+\operatorname{Cos} x)} d x$	3.6 Q-30

Overview

This exercise focuses on evaluating definite integrals, a fundamental concept in calculus. The integrals involve a variety of techniques, such as basic integration rules, substitution, and algebraic manipulation. These problems are designed to build proficiency in solving real-world problems related to areas, rates, and other applications of definite integrals.

Key Concepts

Definite Integrals: The evaluation of an integral over a specific interval.
- Represented as $\int_{a}^{b} f(x) , dx$ , where $a$ and $b$ are the limits of integration.
- Provides the net area under the curve $f(x)$ between $x = a$ and $x = b$ .
Integration Techniques:
- Basic Integration: Applying standard formulas to solve.
- Substitution: Simplifying the integrand by introducing a new variable.
- Algebraic Simplification: Breaking down complex expressions for easier integration.
Special Cases:
- Zero-length intervals yield zero results (e.g., $\int_{1}^{1} f(x) dx = 0$ ).
- Symmetry in definite integrals can simplify calculations.

Important Formulas

Power Rule:
$\int x^n dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$
Basic Integration:
$\int 1 , dx = x + C$
$\int x^n dx = \frac{x^{n+1}}{n+1} + C$
Exponential Functions:
$\int e^x dx = e^x + C$
Substitution Rule:
$\int f(g(x)) g'(x) dx = \int f(u) du$
Definite Integral Evaluation:
$\int_{a}^{b} f(x) dx = F(b) - F(a),$
where $F(x)$ is the antiderivative of $f(x)$ .

Tips and Tricks

Simplify the Integrand: Break down the expression using algebraic techniques before integrating.
Look for Symmetry: For symmetric intervals, odd functions integrate to zero, and even functions double the integral over half the interval.
Check Limits: For substitution, ensure the limits of integration are adjusted for the new variable.
Partial Fractions: Decompose complex fractions into simpler parts when necessary.

Summary

This exercise reinforces the computation of definite integrals using foundational rules and strategies. Key skills include breaking down complex integrals, recognizing patterns, and correctly applying limits to evaluate results. Mastery of these concepts enables solving practical problems in physics, engineering, and economics.

Reference

By Sir Shahzad Sair:

By Great Science Academy: