| --- |
| source: ncert |
| topic: calculus |
| class_level: class_12 |
| chapter: ch7 |
| difficulty: advanced |
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| |
| Class 12 | Ch7: Integrals |
| INDEFINITE INTEGRALS: |
| integral(x^n dx)=x^(n+1)/(n+1)+C (n≠-1). integral(1/x dx)=ln|x|+C. |
| integral(e^x dx)=e^x+C. integral(a^x dx)=a^x/ln(a)+C. |
| integral(sin x dx)=-cos x+C. integral(cos x dx)=sin x+C. |
| integral(tan x dx)=ln|sec x|+C. integral(cot x dx)=ln|sin x|+C. |
| integral(sec x dx)=ln|sec x+tan x|+C. integral(cosec x dx)=-ln|cosec x+cot x|+C. |
| integral(sec^2 x dx)=tan x+C. integral(cosec^2 x dx)=-cot x+C. |
| integral(1/(x^2+a^2) dx)=(1/a)tan^(-1)(x/a)+C. |
| integral(1/sqrt(a^2-x^2) dx)=sin^(-1)(x/a)+C. |
| integral(1/sqrt(x^2+a^2) dx)=ln|x+sqrt(x^2+a^2)|+C. |
| METHODS OF INTEGRATION: |
| METHOD 1 - Substitution: let u=g(x), du=g'(x)dx. |
| integral(2x/(x^2+1) dx): let u=x^2+1, du=2x dx → integral(1/u du)=ln|u|+C=ln|x^2+1|+C. |
| METHOD 2 - Integration by Parts (ILATE rule): |
| integral(u dv)=uv-integral(v du). |
| ILATE order for u: Inverse trig, Logarithm, Algebraic, Trigonometric, Exponential. |
| integral(x e^x dx): u=x, dv=e^x dx. =xe^x-integral(e^x dx)=xe^x-e^x+C=e^x(x-1)+C. |
| integral(x^2 e^x dx): apply by parts TWICE. |
| METHOD 3 - Partial Fractions (for rational functions): |
| (2x+3)/((x-1)(x+2))=A/(x-1)+B/(x+2). Solve for A and B. |
| For repeated factor (x-1)^2: A/(x-1)+B/(x-1)^2+C/(x+2). |
| METHOD 4 - Special Integrals: |
| integral(sin^n x dx) and integral(cos^n x dx): use reduction formulas. |
| integral(sin^2 x dx)=x/2-sin(2x)/4+C. integral(cos^2 x dx)=x/2+sin(2x)/4+C. |
| DEFINITE INTEGRALS: |
| integral[a to b] f(x) dx=F(b)-F(a) (Fundamental Theorem). |
| PROPERTIES: |
| integral[a to b]=- integral[b to a]. |
| integral[a to b] f(x) dx=integral[a to c] f(x) dx+integral[c to b] f(x) dx. |
| integral[0 to a] f(x) dx=integral[0 to a] f(a-x) dx (King's rule). |
| SOLVED EXAMPLES: |
| Example 1: integral(x^2 e^x dx). |
| u=x^2, dv=e^x dx: x^2 e^x-integral(2x e^x dx). |
| Apply by parts again on integral(2x e^x dx): 2[xe^x-e^x]. |
| Answer: x^2 e^x-2xe^x+2e^x+C=e^x(x^2-2x+2)+C. |
| Example 2: integral(dx/(x^2-4))=integral(dx/((x-2)(x+2))). |
| Partial fractions: 1/((x-2)(x+2))=(1/4)(1/(x-2)-1/(x+2)). |
| Answer=(1/4)ln|(x-2)/(x+2)|+C. |
| Example 3: integral(sin^3 x dx)=integral(sin x(1-cos^2 x) dx). |
| Let u=cos x: -integral((1-u^2) du)=-u+u^3/3+C=-cos x+cos^3 x/3+C. |
| COMMON MISTAKES: |
| ILATE: u should be chosen by ILATE order (leftmost=u). |
| Integration by parts may need to be applied multiple times. |
| King's rule: replace x with (a-x) inside integral, keep limits same. |
