mathematics-class-12-unit-07-chapter-10-indefinite-integral-l-3-7-decebustxgq

### <Success style="font-size:25px"> Indefinite Integral L-3 </Success>
### <primary style="font-size:24px"> Problem-1 </primary>
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- $ \int \sin (a x+b) d x=-\frac{\cos (a x+b)}{a}+C$   
- $ a x+b=t$    
- $ \int f(x) d x=F(x)+C$     
- $ \int f(a x+b) d x=\frac{F(a x+b)}{a}+C  \quad \text { (*) }$     
- $ \text {  ose } a x+b=t \Rightarrow a d x=d t$    
- $ d x=\frac{1}{a} d t$     
- $ \int f(t) \frac{d t}{a}=\frac{1}{a} \int f(t) d t=\frac{1}{a} F(t)+C=\frac{F(a x+b)}{a} +C$

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<footer style = "font-size: 18px"> $\rightarrow$ Indefinite integral lecture-10 $\rightarrow$ <span style = "color:blue"> Problem-1 </span> $\rightarrow$ Problem-2 $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Indefinite Integral L-3 </Success>
### <primary style="font-size:24px"> Solution </primary>
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- <ins> case2</ins>

- $ a x+b=t \Rightarrow a d x=d t$       
- $ I=\int \sin t \cos t \frac{d t}{a}$      
- $ =\frac{1}{a} \int \sin t \cos t d t$      
- $ \sin t=u \Rightarrow \cos t d t=d u$      
- $ =\frac{1}{a} \int u d u=\frac{u^2}{2 a}+c$       
- $ =\frac{\sin ^2(a x+b)}{2 a}+c$

- $ \cos 2 \theta=1-2 \sin ^2 \theta$

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<footer style = "font-size: 18px">Problem-1 $\rightarrow$ Problem-2 $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Problem-3 $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Indefinite Integral L-3 </Success>
### <primary style="font-size:24px"> problem-5 </primary>
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- $  I=\int 10^{5^x} \cdot 5^x \cdot d x$   
- $[Put  5^x=t$$ 5^x \log _e 5 d x=d t]$        
- $ =\int 10^t \frac{d t}{\log _e 5}$

- $ =\frac{1}{\log _e 5} \cdot \int 10^t d t=\frac{1}{\log _e{5}} \frac{10^t}{\log _e{10}}+C$  
- $=\frac{1}{\log _e{5}} \frac{10^{5^x}}{\log _e{10}}+C$

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<footer style = "font-size: 18px">Solution $\rightarrow$ Problem-4 $\rightarrow$ <span style = "color:blue"> problem-5 </span> $\rightarrow$ Important trigonometric functions-1 $\rightarrow$ Important trigonometric functions-2 </footer>

### <Success style="font-size:25px"> Indefinite Integral L-3 </Success>
### <primary style="font-size:24px"> Important trigonometric functions-1 </primary>
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- i)

- $ I=\int \tan x d x=\int \frac{\sin x}{\cos x} d x$      
- $ \quad \cos x=t \Rightarrow-\sin x d x=d t$     
- $ I=\int-\frac{d t}{t}=-\log |t|+C$      
- $ =-\log |\cos x|+c=\log |\sec x|+C$   
- $ \int \tan x d x=\log |\sec x|+C$

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<footer style = "font-size: 18px">Problem-4 $\rightarrow$ problem-5 $\rightarrow$ <span style = "color:blue"> Important trigonometric functions-1 </span> $\rightarrow$ Important trigonometric functions-2 $\rightarrow$ Important trigonometric functions-2 </footer>

### <Success style="font-size:25px"> Indefinite Integral L-3 </Success>
### <primary style="font-size:24px"> Important trigonometric functions-2 </primary>
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- ii) $\quad$ $ \int \cot x d x=\log |\sin x|+C $

- $ \text { iii) } \quad$$ I =\int \sec x d x=\int \frac{\sec x(\sec x+\tan x)}{\sec x+\tan x} d x$     
- $ =\int \frac{\sec ^2 x+\sec x \tan x}{\sec x+\tan x} d x$     
- [Put , $\sec x+\tan x=t,$ $ \Rightarrow (\sec^2 x+\sec x.\tan x) d x=dt]$   
- $=\int \frac{d t}{t}$  
- $=\log |t|+C$   $\quad$

- $ {\int \sec x d x=\log |\sec x+\tan x|+C}$

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<footer style = "font-size: 18px">problem-5 $\rightarrow$ Important trigonometric functions-1 $\rightarrow$ <span style = "color:blue"> Important trigonometric functions-2 </span> $\rightarrow$ Important trigonometric functions-2 $\rightarrow$ Problem-6 </footer>

### <Success style="font-size:25px"> Indefinite Integral L-3 </Success>
### <primary style="font-size:24px"> Important trigonometric functions-2 </primary>
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- iv) $\int \operatorname{cosec} x d x$
- $ =\int \frac{\operatorname{cosec} x(\operatorname{cosec} x+\cot x)}{\operatorname{cosec} x+\cot x} d x $

- Choose  
- $\operatorname{cosec} x+\cot x=t$

- $(-\operatorname{cosec} x \cot x-\operatorname{cosec}^2 x) d x=d t$

- $ =-\int \frac{d t}{t}=-\log |t|+C$

- $=\log \left|\frac{1}{\operatorname{cosec} x+\cot x}\right|+C$ $\quad$       
- $[\because \operatorname{cosec}^2 x- {\cot^2 x=1}]$

- $ \int cosec x=\log |\{cosec} x{-\cot x}|+C$

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<footer style = "font-size: 18px">Important trigonometric functions-1 $\rightarrow$ Important trigonometric functions-2 $\rightarrow$ <span style = "color:blue"> Important trigonometric functions-2 </span> $\rightarrow$ Problem-6 $\rightarrow$ Use of trigonometric identities-1 </footer>

### <Success style="font-size:25px"> Indefinite Integral L-3 </Success>
### <primary style="font-size:24px"> Problem-6 </primary>
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- $\int \frac{\sin x}{\sin (x+a)} d x$

- $ x+a=t \Rightarrow d x=d t$

- $ \int \frac{\sin (t-a)}{\sin t} d t=\int\left(\frac{\sin t \cos a-\cos t \sin a}{\sin t}\right) d t$    
- $ =\cos a \int 1 \cdot d t-\sin a \int \cot t d t \quad$    
- $ [\because \int \cot t d t$,$ =\log |\sin t|+c]$      
- $ = \cos a\left(t+c_1\right)-\sin a\left(\log |\sin t|+c_2\right)$    
- $ = \left(x+a+c_1\right) \cos a-\sin a\left(\log |\sin (x+a)|+c_2\right)$    
- $ = x \cos a-\sin a \log |\sin (x+a)|+c$

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<footer style = "font-size: 18px">Important trigonometric functions-2 $\rightarrow$ Important trigonometric functions-2 $\rightarrow$ <span style = "color:blue"> Problem-6 </span> $\rightarrow$ Use of trigonometric identities-1 $\rightarrow$ Use of trigonometric identities-2 </footer>

### <Success style="font-size:25px"> Indefinite Integral L-3 </Success>
### <primary style="font-size:24px"> Use of trigonometric identities-1 </primary>
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- Example:

- $ I=\int \sin ^3 x \cos ^3 x d x$     
- $ =\int \sin ^3 x \cdot \cos ^2 x \cdot \cos x d x,$ $\quad$ ${[ \because \sin ^2 x+ \cos ^2 x=1]}$

- $ =\int \sin ^3 x\left(1-\sin ^2 x\right) \cos x d x$

- $ =\int t^3\left(1-t^2\right) d t$ $ \quad$ $ [Put \sin x=t, \cos x d x=d t]$

- $ =\frac{t^4}{4}-\frac{t^6}{6}+c$    
- $ =\frac{\sin ^4 x}{4}-\frac{\sin ^6 x}{6}+c$

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<footer style = "font-size: 18px">Important trigonometric functions-2 $\rightarrow$ Problem-6 $\rightarrow$ <span style = "color:blue"> Use of trigonometric identities-1 </span> $\rightarrow$ Use of trigonometric identities-2 $\rightarrow$ Use of trigonometric identities-3 </footer>

### <Success style="font-size:25px"> Indefinite Integral L-3 </Success>
### <primary style="font-size:24px"> Use of trigonometric identities-2 </primary>
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- $ I=\int \sin ^3 x \cos ^3 x d x$      
- $ =\int \frac{1}{2^3}(2 \sin x \cos x)^3 d x$      
- $ =\frac{1}{8} \int \sin ^3 2 x d x$    
- $ [Put, 2 x=t, 2 d x=d t, d x=\frac{d t}{2}]$    
- $ =\frac{1}{8} \int \sin ^3 t \frac{d t}{2}$

- $ =\frac{1}{16} \int \sin ^3 t d t$    
- $ =\frac{1}{16} \int \frac{3 \sin t-\sin 3 t}{4} d t$     
- $ [\because \sin 3 x =3 \sin x-4 \sin ^3 x]$     
- $=\frac{1}{64}[3(-\cos t)-(-\frac{\cos 3 t}{3})]+c$      
- $ ={ -\frac{3}{64} cos2x + \frac{1}{64\times 3} cos6x+c}$

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<footer style = "font-size: 18px">Problem-6 $\rightarrow$ Use of trigonometric identities-1 $\rightarrow$ <span style = "color:blue"> Use of trigonometric identities-2 </span> $\rightarrow$ Use of trigonometric identities-3 $\rightarrow$ Thank you </footer>

### <Success style="font-size:25px"> Indefinite Integral L-3 </Success>
### <primary style="font-size:24px"> Use of trigonometric identities-3 </primary>
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- Example: $I=\int \sin 4 x \sin 8 x d x$

- $=\frac{1}{2} \int 2 \sin 4 x \sin 8 x d x$

- $\text { } 2 \sin A \sin B=\cos (A-B)-\cos (A+B)$

- $ I  =\frac{1}{2} \int[\cos (4-8) x-\cos (4+8) x] d x$

- $ =\frac{1}{2} \int(\cos 4 x-\cos 12 x) d x$

- $ ={\frac{1}{2}\left[\frac{\sin 4 x}{4}-\frac{\sin 12 x}{12}\right]+C}$

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<footer style = "font-size: 18px">Use of trigonometric identities-1 $\rightarrow$ Use of trigonometric identities-2 $\rightarrow$ <span style = "color:blue"> Use of trigonometric identities-3 </span> $\rightarrow$ Thank you $\rightarrow$  </footer>