Class 12 Integrals (NCERT)

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### Basic Integration Formulas - **Power Rule:** $\int x^n dx = \frac{x^{n+1}}{n+1} + C$, for $n \neq -1$ - $\int \frac{1}{x} dx = \log|x| + C$ - $\int e^x dx = e^x + C$ - $\int a^x dx = \frac{a^x}{\log a} + C$ - $\int \sin x dx = -\cos x + C$ - $\int \cos x dx = \sin x + C$ - $\int \sec^2 x dx = \tan x + C$ - $\int \csc^2 x dx = -\cot x + C$ - $\int \sec x \tan x dx = \sec x + C$ - $\int \csc x \cot x dx = -\csc x + C$ ### Standard Trigonometric Integrals - $\int \tan x dx = \log|\sec x| + C = -\log|\cos x| + C$ - $\int \cot x dx = \log|\sin x| + C$ - $\int \sec x dx = \log|\sec x + \tan x| + C$ - $\int \csc x dx = \log|\csc x - \cot x| + C$ ### Inverse Trigonometric Integrals - $\int \frac{1}{\sqrt{a^2 - x^2}} dx = \sin^{-1}\left(\frac{x}{a}\right) + C$ - $\int \frac{1}{a^2 + x^2} dx = \frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right) + C$ - $\int \frac{1}{x\sqrt{x^2 - a^2}} dx = \frac{1}{a}\sec^{-1}\left(\frac{x}{a}\right) + C$ ### Methods of Integration #### 1. Integration by Substitution - If $I = \int f(g(x))g'(x) dx$, let $t = g(x)$, then $dt = g'(x) dx$. - So, $I = \int f(t) dt$. #### 2. Integration by Parts - $\int u \, dv = uv - \int v \, du$ - **LIATE Rule:** Choose $u$ in the order: Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential. #### 3. Integration by Partial Fractions - For rational functions $\frac{P(x)}{Q(x)}$, decompose into simpler fractions. - **Types:** - $\frac{px+q}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b}$ - $\frac{px+q}{(x-a)^2} = \frac{A}{x-a} + \frac{B}{(x-a)^2}$ - $\frac{px^2+qx+r}{(x-a)(x^2+bx+c)}$, where $x^2+bx+c$ is irreducible: $\frac{A}{x-a} + \frac{Bx+C}{x^2+bx+c}$ ### Special Integrals - $\int \frac{dx}{x^2 - a^2} = \frac{1}{2a}\log\left|\frac{x-a}{x+a}\right| + C$ - $\int \frac{dx}{a^2 - x^2} = \frac{1}{2a}\log\left|\frac{a+x}{a-x}\right| + C$ - $\int \frac{dx}{\sqrt{x^2 - a^2}} = \log|x + \sqrt{x^2 - a^2}| + C$ - $\int \frac{dx}{\sqrt{x^2 + a^2}} = \log|x + \sqrt{x^2 + a^2}| + C$ - $\int \sqrt{a^2 - x^2} dx = \frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2}\sin^{-1}\left(\frac{x}{a}\right) + C$ - $\int \sqrt{x^2 - a^2} dx = \frac{x}{2}\sqrt{x^2 - a^2} - \frac{a^2}{2}\log|x + \sqrt{x^2 - a^2}| + C$ - $\int \sqrt{x^2 + a^2} dx = \frac{x}{2}\sqrt{x^2 + a^2} + \frac{a^2}{2}\log|x + \sqrt{x^2 + a^2}| + C$ ### Definite Integrals - **Definition:** $\int_a^b f(x) dx = F(b) - F(a)$, where $F(x)$ is the antiderivative of $f(x)$. #### Properties of Definite Integrals 1. $\int_a^b f(x) dx = -\int_b^a f(x) dx$ 2. $\int_a^a f(x) dx = 0$ 3. $\int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx$ 4. $\int_a^b f(x) dx = \int_a^b f(a+b-x) dx$ 5. $\int_0^a f(x) dx = \int_0^a f(a-x) dx$ 6. $\int_0^{2a} f(x) dx = \int_0^a f(x) dx + \int_0^a f(2a-x) dx$ 7. $\int_0^{2a} f(x) dx = 2\int_0^a f(x) dx$, if $f(2a-x) = f(x)$ 8. $\int_0^{2a} f(x) dx = 0$, if $f(2a-x) = -f(x)$ 9. $\int_{-a}^a f(x) dx = 2\int_0^a f(x) dx$, if $f(x)$ is an even function ($f(-x)=f(x)$) 10. $\int_{-a}^a f(x) dx = 0$, if $f(x)$ is an odd function ($f(-x)=-f(x)$) ### Applications of Integrals #### Area Under Simple Curves - Area bounded by $y = f(x)$, $x$-axis, $x=a$, and $x=b$: $\int_a^b y \, dx$ or $\int_a^b f(x) dx$ - Area bounded by $x = g(y)$, $y$-axis, $y=c$, and $y=d$: $\int_c^d x \, dy$ or $\int_c^d g(y) dy$ #### Area Between Two Curves - Area bounded by $y = f(x)$ and $y = g(x)$ from $x=a$ to $x=b$: $\int_a^b |f(x) - g(x)| dx$ - If $f(x) \ge g(x)$ on $[a,b]$, then Area $= \int_a^b (f(x) - g(x)) dx$