Integrals — Class 12

Introduction

English: This chapter covers integrals — the reverse process of differentiation. You will master indefinite integrals (antiderivatives), definite integrals and their properties, Fundamental Theorem of Calculus, techniques (substitution, parts, partial fractions, trigonometric integrals), and applications (area, average value, physics). Clear proofs, diagrams, 8 step-by-step solved examples, and a thorough practice set are provided. SEO-rich keywords: Class 12 Maths, integrals notes, definite integral, indefinite integral, FTC, rsetu.link.

Marathi / मराठी (संक्षेप): हे chapter integrals ची सखोल समज देते: antiderivatives, definite integrals, FTC, विविध integration पद्धती आणि उपयोग. सर्व सिद्धांत व उदाहरणे सोप्या टप्प्यात समजावून दिली आहेत.

Basic Concepts & Definitions

1. Indefinite integral (antiderivative): \(\int f(x)\,dx=F(x)+C\) where \(F'(x)=f(x)\).
2. Definite integral: \(\int_a^b f(x)\,dx\) interpreted as signed area; properties: additivity, linearity, reversal sign.
3. Fundamental Theorem of Calculus (FTC): Part 1: If \(F(x)=\int_a^x f(t)dt\) then \(F'(x)=f(x)\). Part 2: \(\int_a^b f(x)dx = F(b)-F(a)\) where F is any antiderivative.

Techniques of Integration (with when to use)

• Substitution (u-sub): when integrand is composite, e.g. \(\int 2x\cos(x^2)dx\).
• Integration by parts: \(\int u\,dv = uv - \int v\,du\) — useful for polynomial×log/trig/exponential.
• Partial fractions: rational functions where deg(num)<deg(denom) after division.
• Trigonometric identities & reduction: use for \(\sin^n x, \cos^n x\) integrals.
• Trigonometric substitution for √(a^2 - x^2), √(a^2 + x^2), √(x^2 - a^2).

Important Formulas & Identities

• \(\int x^n dx = \frac{x^{n+1}}{n+1} + C, n\neq -1\)
• \(\int e^{ax}dx = \frac{1}{a}e^{ax} + C\)
• \(\int \sin x dx = -\cos x + C, \int \cos x dx = \sin x + C\)
• Integration by parts: \(\int u\,dv = uv - \int v\,du\)
• Partial fractions decomposition basics for linear/quadratic factors.

Solved Examples (8) — step-by-step

1. Example 1: Evaluate \(\int 2x\cos(x^2)dx\).

   Let u = x^2 ⇒ du = 2x dx ⇒ integral = \(\int \cos u\,du = \sin u + C = \sin(x^2)+C\).

2. Example 2: Compute \(\int_0^{\pi} x\sin x\,dx\) using integration by parts.

   Let u=x, dv=sin x dx ⇒ du=dx, v=-cos x. So \(uv|_0^{\pi} - \int_0^{\pi} v du = [-x\cos x]_0^{\pi} + \int_0^{\pi} \cos x dx = [-\pi(-1)-0] + [\sin x]_0^{\pi} = \pi\).

3. Example 3: Evaluate \(\int \frac{dx}{x^2-1}\) by partial fractions.

   Decompose: \(\frac{1}{x^2-1}=\frac{1/2}{x-1} - \frac{1/2}{x+1}\). Integrate → \(\frac{1}{2}\ln|x-1| - \frac{1}{2}\ln|x+1| + C\).

4. Example 4: Calculate area between y=x^2 and y=x from x=0 to 1.

   Area = \(\int_0^1 (x - x^2) dx = [\tfrac{x^2}{2} - \tfrac{x^3}{3}]_0^1 = 1/2 - 1/3 = 1/6\).

5. Example 5: Evaluate \(\int x e^{2x} dx\) by parts.

   u = x, dv = e^{2x} dx ⇒ du=dx, v= e^{2x}/2. So result = x e^{2x}/2 - \int e^{2x}/2 dx = x e^{2x}/2 - e^{2x}/4 + C = e^{2x}(2x-1)/4 + C.

6. Example 6: Compute \(\int \sin^2 x dx\) using identity.

   Use \(\sin^2 x = (1-\cos 2x)/2\). Integral = \(\tfrac{x}{2} - \tfrac{\sin 2x}{4} + C\).

7. Example 7: Evaluate improper integral \(\int_1^{\infty} \frac{1}{x^2} dx\).

   Compute limit: \(\lim_{b\to\infty} [-1/x]_1^{b} = 1\).

8. Example 8: Use substitution to evaluate \(\int \frac{dx}{\sqrt{a^2 - x^2}}\).

   Let x = a sin θ ⇒ dx = a cos θ dθ; integral → \(\int dθ = \sin^{-1}(x/a) + C\).

Applications of Integrals

• Area between curves, area in polar coordinates (advanced)
• Volume of revolution (disc/washer method)
• Center of mass, work done by variable force
• Average value of a function: \(f_{avg}=\frac{1}{b-a}\int_a^b f(x)dx\)

Practice Questions (12)

1. Evaluate \(\int (3x^2 - 2x +1)dx\).
2. Find \(\int_0^1 x e^{x^2} dx\).
3. Integrate \(\int \frac{dx}{x^2+4}\).
4. Compute area between y=sin x and y=0 from 0 to π.
5. Use substitution to evaluate \(\int \frac{x}{\sqrt{1-x^2}}dx\).
6. Evaluate \(\int x^2\ln x dx\) (by parts).
7. Determine convergence of \(\int_0^1 \frac{dx}{\sqrt{x}}\).
8. Compute \(\int \frac{dx}{(x+1)(x^2+1)}\) using partial fractions.
9. Find area enclosed by y=x^3 and y=x from x=0 to 1.
10. Evaluate improper integral \(\int_0^{\infty} e^{-x} dx\).
11. Show that derivative of \(F(x)=\int_0^{x^2} \sin t dt\) is \(F'(x)=2x\sin(x^2)\).
12. Find average value of f(x)=x^2 on [0,2].

Summary Table (Quick Revision)

Answer Key — Practice Qs (brief)

1. \(x^3 - x^2 + x + C\).
2. Let u=x^2 ⇒ du=2x dx ⇒ integral = \(\tfrac{1}{2}\int_0^1 e^{u} du = \tfrac{1}{2}(e-1)\).
3. \(\frac{1}{2}\tan^{-1}(x/2) + C\).
4. Area = \(\int_0^{\pi} \sin x dx = 2\).
5. Use x=sin θ substitution → result = -√(1-x^2) + C (or trig substitution steps shown).
6. By parts: u=ln x, dv = x^2 dx ⇒ result = \(\tfrac{x^3}{3}\ln x - \tfrac{x^3}{9} + C\).
7. Convergent; integral = 2.
8. Partial fraction decomposition (steps lead to logs and arctan).
9. Area = \(\int_0^1 (x - x^3) dx = \tfrac{1}{2} - \tfrac{1}{4} = \tfrac{1}{4}\).
10. 1 (compute limit of integral) — convergent; value =1.
11. Use chain rule + FTC: F'(x)= (d/dx)(x^2) * sin(x^2) = 2x sin(x^2).
12. Average = \(\tfrac{1}{2} \int_0^2 x^2 dx = \tfrac{1}{2} [\tfrac{x^3}{3}]_0^2 = \tfrac{4}{3}\).