mathematics-class-12-unit-07-chapter-13-indefinite-integral-l-6-7-rnyj9fvvtvw

### <Success style="font-size:25px"> Indefinite Integral L-6 </Success>
### <primary style="font-size:24px"> Problem-1 </primary>
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- $\int {\underset{I}x^2 \underset{II}e^x} d x$

- Solution:

- $=x^2\left(\int e^x d x\right)-\int 2 x\left(\int e^x d x\right) d x $

- $ =x^2 e^x - 2 \int x e^x d x $

- $ =x^2 e^x - 2\left[x\left(\int e^x d x\right)-\int 1\left(\int e^x d x\right) d x\right] $

- $ =x^2 e^x-2\left[x e^x-\int e^x d x\right] $

- $\int x^2 e^x=x^2 e^x-2 x e^x+2 e^x+C$

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<footer style = "font-size: 18px">Indefinite integral lecture-13 $\rightarrow$ Method of Integration by parts $\rightarrow$ <span style = "color:blue"> Problem-1 </span> $\rightarrow$ Problem-2 $\rightarrow$ Problem-3 </footer>

### <Success style="font-size:25px"> Indefinite Integral L-6 </Success>
### <primary style="font-size:24px"> Problem-2 </primary>
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- $\int \underset{I}x \underset{II}\sin 3 x d x$

- Solution:

- $= x \int \sin 3 x d x-\int 1 \cdot(\int\sin 3 x d x)d x $

- $=x\left(\frac{-\cos 3 x}{3}\right)-\int\left(-\frac{\cos 3 x}{3}\right) d x $

- $[\int f(a x+b) d x = \frac{F(ax+b)}{a}+c$
- $\quad if \int f(x)dx = F(x)+c ]$

- $ =-\frac{x \cos 3 x}{3}+\frac{1}{3} \int \cos 3 x d x$
- $ =-\frac{x \cos 3 x}{3}+\frac{1}{3}\left[\frac{\sin 3 x}{3}\right]+c$

- $\int x \sin 3 x=-x \frac{\cos 3 x}{3}+\frac{1}{9} \sin 3 x+c  $

- $  \int \underset{I}f(x) \underset{II}g(x) d x $ 
- $ = \underset{I}f(x). \int \underset{II}g(x) d x-\int \underset{I}f^{\prime}(x) (\int \underset{II}g(x) d x) d x$

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

<h3 id="success-stylefont-size25px-indefinite-integral-l-6-success"><Success style="font-size:25px"> Indefinite Integral L-6 </Success></h3>
<h3 id="primary-stylefont-size24px-problem-3-primary"><primary style="font-size:24px"> Problem-3 </primary></h3>
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<ul>
<li>
<p>$I=\int {x} \log x d x$</p>
</li>
<li>
<p>Solution:</p>
</li>
<li>
<p>$ I=\int \underset{I}{x} \underset{II}\log x d x $</p>
</li>
<li>
<p>$=x\left(\int \log x d x\right)- \int 1 \cdot\left(\int \log x\right) d x $</p>
</li>
<li>
<p>$ =x {\int \log x d x}-{\int\left(\int \log x d x\right) d x}$</p>
</li>
<li>
<p>$ \int \log x d x=\text { ? }$</p>
</li>
<li>
<p>$I=\int \underset{II}{x} \underset{I}\log x d x$</p>
</li>
<li>
<p>$ =\log x \cdot \int x d x-$</p>
</li>
<li>
<p>$ \int \frac{1}{x} \times\left(\int x d x\right) d x$</p>
</li>
</ul>
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<hr>
<footer style = "font-size: 18px">Problem-1 $\rightarrow$ Problem-2 $\rightarrow$ <span style = "color:blue"> Problem-3 </span> $\rightarrow$ Solution $\rightarrow$ Problem-4 </footer>

<h3 id="success-stylefont-size25px-indefinite-integral-l-6-success-1"><Success style="font-size:25px"> Indefinite Integral L-6 </Success></h3>
<h3 id="primary-stylefont-size24px-solution-primary"><primary style="font-size:24px"> Solution </primary></h3>
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<main>
<ul>
<li>
<p>$ =\log x \cdot \frac{x^2}{2}-\int \frac{1}{x} \times \frac{x^2}{2} d x $</p>
</li>
<li>
<p>$ =\frac{x^2}{2} \log x-\frac{1}{2} \int x d x$</p>
</li>
<li>
<p>$ =\frac{x^2}{2} \log x-\frac{1}{2} \times \frac{x^2}{2}+c$</p>
</li>
<li>
<p>$ I=\frac{x^2}{2} \log x-\frac{x^2}{4}+c$</p>
</li>
</ul>
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<hr>
<footer style = "font-size: 18px">Problem-2 $\rightarrow$ Problem-3 $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Problem-4 $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Indefinite Integral L-6 </Success>
### <primary style="font-size:24px"> Problem-4 </primary>
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<main>

- I=$\int \underset{II}{x} \underset{I}{\sin ^{-1} x} d x$

- Solution:
- $ =\sin ^{-1} x \cdot \frac{x^2}{2}-\int \frac{1}{\sqrt{1-x^2}} \cdot \frac{x^2}{2} d x $

- $ =\frac{x^2 \sin ^{-1} x}{2}-\frac{1}{2} \int \frac{x^2 d x}{\sqrt{1-x^2}}$

- $ =\frac{x^2 \sin ^{-1} x}{2}-\frac{1}{2} \int \frac{-(1-x^2)+1}{\sqrt{1-x^2}} d x$

- $=\frac{x^2 \sin ^{-1}x}{2}$+$\frac{1}{2} \int \sqrt{1-x^2} d x$ - $\frac{1}{2} \int \frac{1 dx}{\sqrt{1-x^2}}$

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<footer style = "font-size: 18px">Problem-3 $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Problem-4 </span> $\rightarrow$ Solution $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Indefinite Integral L-6 </Success>
### <primary style="font-size:24px"> Solution </primary>
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<main>

- $I_1 =\int {\cos \theta}.{\cos \theta d \theta} $

- $ [\cos 2 \theta=2 \cos ^2 \theta-1]$

- $ [\cos ^2 \theta=\frac{\cos 2 \theta-1}{2}]$

- $I_1 =\int \cos ^2 \theta d\theta $

- $ =\frac{1}{2} \int(\cos 2 \theta-1) d\theta $

- $ =\frac{1}{2}(\frac{\sin 2 \theta}{2}-\theta)$

- $=\frac{1}{4} \sin 2\left(\sin ^{-1} x\right)-\frac{1}{2} \sin ^{-1} x $

</main>
<hr>
<footer style = "font-size: 18px">Problem-4 $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Solution $\rightarrow$ Choice of first & second function </footer>

### <Success style="font-size:25px"> Indefinite Integral L-6 </Success>
### <primary style="font-size:24px"> Solution </primary>
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<main>

- $ I=\frac{x^2 \sin ^{-1} x}{2}-\frac{1}{2}.(\frac{2x}{4} \sqrt{1-x^2}-\frac{1}{2} \sin ^{-1} x) -\frac{1}{2} \sin ^{-1} x+c $

- $ I=\frac{x^2 \sin ^{-1} x}{2}-\frac{1}{2}.[\frac{x}{2} \sqrt{1-x^2}-\frac{1}{2} \sin ^{-1} x] -\frac{1}{2} \sin ^{-1} x+c $

- $ \left[\because \sin 2 \theta=2 \sin\theta \cos \theta\  =2 x.\sqrt{1-\sin ^2 \theta}  =2 x \sqrt{1-x^2}\right]$

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<footer style = "font-size: 18px">Solution $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Choice of first & second function $\rightarrow$ Problem-5 </footer>

### <Success style="font-size:25px"> Indefinite Integral L-6 </Success>
### <primary style="font-size:24px"> Choice of first & second function </primary>
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<main>

- Inverse trigonometric

- Log function

- Algebraic

- Trigonometric

- Exponential

- $\int \underset{II}x \underset{I}{logx} dx$

- $  \int \underset{I}f(x) \underset{II}g(x) d x $
- $ = \underset{I}f(x). \int \underset{II}g(x) d x-\int \underset{I}f^{\prime}(x) (\int \underset{II}g(x) d x) d x$

</main>
<hr>
<footer style = "font-size: 18px">Solution $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Choice of first & second function </span> $\rightarrow$ Problem-5 $\rightarrow$ Problem-6 </footer>

### <Success style="font-size:25px"> Indefinite Integral L-6 </Success>
### <primary style="font-size:24px"> Problem-6 </primary>
<hr>
<main>

- $I =\int \underset{I}x \underset{II}\log x d x $

- Solution
- $ =x \int \log x d x-\int 1 \cdot\left(\int \log x d x\right) d x$

- $ =x(x \log x-x)-\int(x \log x-x) d x $

- $ =x^2 \log x-x^2-\int x \log x d x+\frac{x^2}{2}+C $

- $ I  =x^2 \log x-\frac{x^2}{2}-I+C $

- $ I  =\frac{x^2 \log x}{2}-\frac{x^2}{4}+C$

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<hr>
<footer style = "font-size: 18px">Choice of first & second function $\rightarrow$ Problem-5 $\rightarrow$ <span style = "color:blue"> Problem-6 </span> $\rightarrow$ Problem-7 $\rightarrow$ Solution </footer>

<h3 id="success-stylefont-size25px-indefinite-integral-l-6-success-2"><Success style="font-size:25px"> Indefinite Integral L-6 </Success></h3>
<h3 id="primary-stylefont-size24px-solution-primary-1"><primary style="font-size:24px"> Solution </primary></h3>
<hr>
<main>
<div style='float: left; width:100%;font-size:30px;text-align:left;'>
<ul>
<li>
<p>[Put $1+x^2=t \Rightarrow 2 x d x=d t \Rightarrow x d x=\frac{d t}{2}$]</p>
</li>
<li>
<p>$=x \tan ^{-1} x-\frac{1}{2} \int \frac{d t}{t}$</p>
</li>
<li>
<p>$=x \tan ^{-1} x-\frac{1}{2} \log |t|+C$</p>
</li>
<li>
<p>$ I=x \tan ^{-1} x-\frac{1}{2} \log \left|1+x^2\right|+C $</p>
</li>
</ul>


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<div style='float: right; width:50%;'>


</main>
<hr>
<footer style = "font-size: 18px">Problem-6 $\rightarrow$ Problem-7 $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Problem-8 $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Indefinite Integral L-6 </Success>
### <primary style="font-size:24px"> Solution </primary>
<hr>
<main>

- $I_2=\frac{e^{m x} \sin n x}{m}-\frac{n}{m^2} e^{m x} \cos n x \\  -\frac{n^2}{m^2} {\int e^{m x} \sin n x d x}$

- [$\because I_2 = \int e^{m x} \sin n x d x$ ]

- $\Rightarrow I_2\left(1+\frac{n^2}{m^2}\right)=\frac{m e^{m x} \sin n x-n e^{m x} \cos n x}{m^2}$

- $\Rightarrow I_2=\int e^{m x} \sin n x d x=\frac{1}{m^2+n^2}[m \sin n x-n \cos nx] e^{m x}$

</main>
<hr>
<footer style = "font-size: 18px">Solution $\rightarrow$ Problem-8 $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Special function $\rightarrow$ Special function </footer>

### <Success style="font-size:25px"> Indefinite Integral L-6 </Success>
### <primary style="font-size:24px"> Special function </primary>
<hr>
<main>

- $I  =\frac{x}{2}\sqrt{x^2-a^2}-\frac{a^2}{2} \log\left|x+\sqrt{x^2-a^2}\right|+c$

- 1. $\int \sqrt{x^2-a^2} d x=\frac{x}{2} \sqrt{x^2-a^2}-\frac{a^2}{2} \log
\left|x+\sqrt{x^2-a^2}\right|+c$

- 2. $\int \sqrt{a^2-x^2} d x=\frac{x}{2} \sqrt{a^2-x^2}+\frac{a^2}{2} \sin ^{-1} \frac{x}{a}+c$

- 3. $\int \sqrt{x^2+a^2} d x=\frac{x}{2} \sqrt{x^2+a^2}+\frac{a^2}{2} \log \left|x+\sqrt{x^2+a^2}\right|+c$

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<div style='float: right; width:50%;'>

</main>
<hr>
<footer style = "font-size: 18px">Solution $\rightarrow$ Special function $\rightarrow$ <span style = "color:blue"> Special function </span> $\rightarrow$ Problem-9 $\rightarrow$ Thankyou </footer>

### <Success style="font-size:25px"> Indefinite Integral L-6 </Success>
### <primary style="font-size:24px"> Problem-9 </primary>
<hr>
<main>

- $I=\int \sqrt{1-4 x-x^2} d x$

- Solution:
- [$ 1-4 x-x^2=1-(4 x+x^2+4 -4) = 5-(x+2)^2 $]

- $I= \int \sqrt{5-(x+2)^2} d x $

- [$Put, x+2=t \Rightarrow dx=dt $ ]

- $=  \int \sqrt{5-t^2} d t $

- $=  \frac{t}{2} \sqrt{5-t^2}+\frac{5}{2} \sin ^{-1} \frac{t}{\sqrt{5}}+C $

- $=  \frac{x+2}{2} \sqrt{1-4 x+x^2}+\frac{5}{2} \sin ^{-1} \frac{x+2}{\sqrt{5}}+C$