mathematics-class-12-unit-07-chapter-09-indefinite-integral-l-2-7-0qqmlui2e80

### <Success style="font-size:25px"> Indefinite Integral L-2 </Success>
### <primary style="font-size:24px"> Property-1 of indefinite integrals </primary>
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- 1. a) $\frac{d}{d x} \int f(x) d x=f(x)$

- b) $\int f^{\prime}(x) d x=f(x)+c, c \in R$.

- Proof:
- (a) $ ~ \frac{d}{d x}(F(x))=f(x) \Rightarrow \int f(x) d x=F(x)+c$

- $\frac{d}{d x}\left[\int f(x) d x\right]=\frac{d}{d x}[F(x)+c]=\frac{d F}{d x}+0$

- $\frac{d}{d x} \int f(x) d x=f(x) \text {. }$

- (b) $ ~  {f^{\prime}(x)}=\frac{d}{d x}{f(x)} \Rightarrow \int f^{\prime}(x) d x=f(x)+c , c\in \mathbb{R}$

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<footer style = "font-size: 18px">Indefinite integral lecture-9 $\rightarrow$ Problem-1 $\rightarrow$ <span style = "color:blue"> Property-1 of indefinite integrals </span> $\rightarrow$ Property-2 of indefinite integrals $\rightarrow$ Linearity property </footer>

### <Success style="font-size:25px"> Indefinite Integral L-2 </Success>
### <primary style="font-size:24px"> Property-2 of indefinite integrals </primary>
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- 2. $\frac{d}{d x}\left(\int f(x) d x\right)=\frac{d}{d x}\left(\int g(x) d x\right)$

- $\Rightarrow  \frac{d}{d x}\left[\int f(x) d x-\int g(x) d x\right]=0$

- $\Rightarrow \int f(x) d x-\int g(x) d x=C$

- $\\{\int g(x) d x+C_1, C_1 \in \mathbb{R}\\}$ & $\\{\int f(x) dx+C_2, C_2 \in \mathbb{R}\\\}$

- $\int f(x) d x=\int g(x) d x .$

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<footer style = "font-size: 18px">Problem-1 $\rightarrow$ Property-1 of indefinite integrals $\rightarrow$ <span style = "color:blue"> Property-2 of indefinite integrals </span> $\rightarrow$ Linearity property $\rightarrow$ Scalar multiplication </footer>

### <Success style="font-size:25px"> Indefinite Integral L-2 </Success>
### <primary style="font-size:24px"> Linearity property </primary>
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- $\int[f(x)+g(x)] d x=\int f(x) d x+\int g(x) d x $

- Proof: $\frac{d}{d x}\left[\int(f(x)+g(x)) d x\right]$

- =f(x)+g(x)......(i)

- $\frac{d}{d x}\left[\int f(x) d x+\int g(x) d x\right]=\frac{d}{d x} \int f(x) d x+\frac{d}{d x} \int g(x) d x$

- $=f(x)+g(x) \ldots \text { (ii) }$

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<footer style = "font-size: 18px">Property-1 of indefinite integrals $\rightarrow$ Property-2 of indefinite integrals $\rightarrow$ <span style = "color:blue"> Linearity property </span> $\rightarrow$ Scalar multiplication $\rightarrow$ Scalar multiplication </footer>

### <Success style="font-size:25px"> Indefinite Integral L-2 </Success>
### <primary style="font-size:24px"> Scalar multiplication </primary>
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- $\int k f(x) d x=k \int f(x) d x \quad k \rightarrow \text { constant }$

- $\frac{d}{d x} \int k f(x) d x=k f(x)$

- $\frac{d}{d x}\left[k \int f(x) d x\right] =k \frac{d}{d x} \int f(x) d x =k f(x)$

- $\int k f(x) d x  =k \int f(x) d x$

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<footer style = "font-size: 18px">Property-2 of indefinite integrals $\rightarrow$ Linearity property $\rightarrow$ <span style = "color:blue"> Scalar multiplication </span> $\rightarrow$ Scalar multiplication $\rightarrow$ Some important formulae-1 </footer>

### <Success style="font-size:25px"> Indefinite Integral L-2 </Success>
### <primary style="font-size:24px"> Scalar multiplication </primary>
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- for constants $k_1, k_3 \ldots k_n \ $ functions;

- $f_1(x), f_2(x), \ldots f_n(x)$,

- $\int\left[k_1 f_1(x)\right.  \left.+k_2 f_2(x)+\cdots+k_n f_n(x)\right] d x$

- $= k_1 \int f_1(x) d x+k_2 \int f_2(x) d x+\cdots+  +k_n \int f_n(x) d x $

- Example $I=\int\left(a x^2+b x+c\right) d x$

- $ =a \int x^2+b \int x d x+c \cdot \int 1 \cdot d x $

- $I  =a \frac{x^3}{3}+b \frac{x^2}{2}+c x+c_1$

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<footer style = "font-size: 18px">Linearity property $\rightarrow$ Scalar multiplication $\rightarrow$ <span style = "color:blue"> Scalar multiplication </span> $\rightarrow$ Some important formulae-1 $\rightarrow$ Some important formulae-2 </footer>

### <Success style="font-size:25px"> Indefinite Integral L-2 </Success>
### <primary style="font-size:24px"> Some important formulae-1 </primary>
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- $  \frac{d}{d x}(\frac{x^{n+1}}{n+1})=x^n ;\quad \int x^n d x=\frac{x^{n+1}}{n+1}+c, n \neq-1$

- $ \frac{d}{d x}(x)=1 ;\quad\int 1 \cdot d x=x+C$

- $ \frac{d}{d x}(\sin x)=\cos x ; \quad\int \cos x d x=\sin x+C$

- $  \frac{d}{d x}(-\cos x)=\sin x ;\quad \int \sin x d x=-\cos x+C$

- $  \frac{d}{d x}(\tan x)=\sec ^2 x;\quad \int \sec ^2 x d x=\tan x+C$

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<footer style = "font-size: 18px">Scalar multiplication $\rightarrow$ Scalar multiplication $\rightarrow$ <span style = "color:blue"> Some important formulae-1 </span> $\rightarrow$ Some important formulae-2 $\rightarrow$ Some important formulae-3 </footer>

### <Success style="font-size:25px"> Indefinite Integral L-2 </Success>
### <primary style="font-size:24px"> Some important formulae-2 </primary>
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- $ \frac{d}{d x}(-\cot x)=\operatorname{cosec}^2 x ;\quad \int \operatorname{cosec}^2 x d x=-\cot x+C$

- $ \frac{d}{d x}(\sec x)=\sec x \tan x  ;\quad\int \sec x \tan x d x=\sec x+C$

- $ \frac{d}{d x}(-\operatorname{cosec} x)=\operatorname{cosec} x \cot x ;\quad \int \operatorname{cosec} x \cot x dx=\operatorname{-cosec} x+C$

- $ \frac{d}{d x}\left(\sin ^{-1} x\right)=\frac{1}{\sqrt{1-x^2}} ;\quad \int \frac{1}{\sqrt{1-x^2}} d x=\sin ^{-1} x+C$

- $ \frac{d}{d x}\left(-\cos ^{-1} x\right)=\frac{1}{\sqrt{1-x^2}} ;\quad \int \frac{1}{\sqrt{1-x^2}} d x=-\cos ^{-1} x+C$

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<footer style = "font-size: 18px">Scalar multiplication $\rightarrow$ Some important formulae-1 $\rightarrow$ <span style = "color:blue"> Some important formulae-2 </span> $\rightarrow$ Some important formulae-3 $\rightarrow$ Some important formulae-4 </footer>

### <Success style="font-size:25px"> Indefinite Integral L-2 </Success>
### <primary style="font-size:24px"> Some important formulae-3 </primary>
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- $ \\frac{d}{d x}\left(\tan ^{-1} x\right)=\frac{1}{1+x^2} ;\quad  \int \frac{1}{1+x^2} d x=\tan ^{-1} x+C$

- $ \\frac{d}{d x}\left(-\cot ^{-1} x\right)=\frac{1}{1+x^2} ;\quad  \int \frac{d x}{1+x^2}=-\cot ^{-1} x+C$

- $ \\frac{d}{d x}\left(\sec ^{-1} x\right)=\frac{1}{x \sqrt{x^2-1}} ;\quad  \int \frac{d x}{x \sqrt{x^2-1}}=\sec ^{-1} x+C $

- $ \\frac{d}{d x}\left(-\operatorname{cosec}^{-1} x\right)=\frac{1}{x \sqrt{x-1}} ;\quad  \int \frac{d x}{x
\sqrt{x^2-1}}=-\operatorname{cosec}^{-1} x + C $

- $ \\frac{d}{d x}\left(e^x\right)=e^x ;\quad \int e^x d x=e^x + C$

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<footer style = "font-size: 18px">Some important formulae-1 $\rightarrow$ Some important formulae-2 $\rightarrow$ <span style = "color:blue"> Some important formulae-3 </span> $\rightarrow$ Some important formulae-4 $\rightarrow$ Example-1 </footer>

### <Success style="font-size:25px"> Indefinite Integral L-2 </Success>
### <primary style="font-size:24px"> Example-4 </primary>
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- $\int \frac{\sec ^2 x}{\operatorname{cosec}^2 x} d x $   
- $=  \int \frac{1 / \cos ^2 x}{1 / \sin ^2 x} d x $    
- $=  \int \frac{\sin ^2 x}{\cos ^2 x} d x $  
- $=  \int \tan ^2 x d x $   
- $\quad$ $(1+\tan ^2 x =\sec ^2 x)$   
- $=  \int\left(\sec ^2 x-1\right) d x$  
- $= \tan x-x+c$

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<footer style = "font-size: 18px">Example-1 $\rightarrow$ Example-3 $\rightarrow$ <span style = "color:blue"> Example-4 </span> $\rightarrow$ Comparison of differentiation & Int $\rightarrow$ Method by substitution </footer>

### <Success style="font-size:25px"> Indefinite Integral L-2 </Success>
### <primary style="font-size:24px"> Comparison of differentiation & Int </primary>
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|Diff      | Int. |
| -------- | -------- | 
| Operator   |Operator    |
| $ \frac{d}{d x} f(x)=f^{\prime}(x)$   |$ \int f(x) d x=F(x)+c $   |
| Linearity Prop.| Linearity Prop   |
|Unique  |Unique upto a constant|
| $ \frac{d y}{d x}_{(x_0, y_0)}$  |No such meaning   |
| Limiting process| Limiting process   |

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<footer style = "font-size: 18px">Example-3 $\rightarrow$ Example-4 $\rightarrow$ <span style = "color:blue"> Comparison of differentiation & Int </span> $\rightarrow$ Method by substitution $\rightarrow$ Example </footer>