mathematics-class-12-unit-07-chapter-08-indefinite-integral-l-1-7-8fn5hsybmfu

### <Success style="font-size:25px"> Indefinite Integral L-1 </Success>
### <primary style="font-size:24px"> Slope of tangent </primary>
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<footer style = "font-size: 18px"> $\rightarrow$ Indefinite integral lecture-8 $\rightarrow$ <span style = "color:blue"> Slope of tangent </span> $\rightarrow$ Area under curve $\rightarrow$ Area under curve </footer>

### <Success style="font-size:25px"> Indefinite Integral L-1 </Success>
### <primary style="font-size:24px"> Area under curve </primary>
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- 1. Assume that $f(x), x \in[a, b]$ is continuous & differentiable on $(a, b)$ so that $f^{\prime}(x)$ is known.
- Given $f^{\prime}(x)$, can we find the $f(x)$?

- 2. Assume that $f(x)$ is a continuous function on $[a, b]$,     
- Area A ?

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<footer style = "font-size: 18px">Indefinite integral lecture-8 $\rightarrow$ Slope of tangent $\rightarrow$ <span style = "color:blue"> Area under curve </span> $\rightarrow$ Area under curve $\rightarrow$ First Fundamental theorem of calculus </footer>

### <Success style="font-size:25px"> Indefinite Integral L-1 </Success>
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- $ f(x)=x, [0,a] \quad a>0$

- $ A(x) \quad 0<x<a  $ 
- $ A(x)=\frac{1}{2}$ $\times$$x$$\times$$x$     
- $ A(x)=\frac{1}{2} x^2 $    
- $ A(a)=\frac{1}{2} a^2,  A(0)=0 $

- $ \frac{d}{d x}(A(x))=\frac{d}{d x}\left(\frac{x^2}{2}\right)=\frac{2 x}{2}=x $

- $ \frac{d}{d x} (A(x)) = x $

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<footer style = "font-size: 18px">Slope of tangent $\rightarrow$ Area under curve $\rightarrow$ <span style = "color:blue"> Area under curve </span> $\rightarrow$ First Fundamental theorem of calculus $\rightarrow$ Antiderivative </footer>

### <Success style="font-size:25px"> Indefinite Integral L-1 </Success>
### <primary style="font-size:24px"> Antiderivative </primary>
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- $ \frac{d}{d x}(\sin x)=\cos x $    
- $ \frac{d}{d x}\left(\frac{e^{n x}}{n}\right)=e^{n x}, n>0 $    
- $ \frac{d}{d x}(\tan x)=\sec ^2 x $

- $ \sin x- \text { anti-derivative of } \cos x $    
- $ \frac{e^{n x}}{n}-\text { anti-derivative of } e^{n x} $   
- $ \tan x-\text { anti-derivative of } \sec ^2 x $

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<footer style = "font-size: 18px">Area under curve $\rightarrow$ First Fundamental theorem of calculus $\rightarrow$ <span style = "color:blue"> Antiderivative </span> $\rightarrow$ Antiderivative $\rightarrow$ Antiderivative </footer>

### <Success style="font-size:25px"> Indefinite Integral L-1 </Success>
### <primary style="font-size:24px"> Antiderivative </primary>
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- $ \frac{d}{d x}(\sin x+1)=\frac{d}{d x} \sin x+\frac{d}{d x} 1 $

- $ =\frac{d}{d x} \sin x=\cos x $

- $\cos x$ $\quad$ has $\quad$ $(\sin x+1)$  also as antiderivative.

- $\frac{d}{d x}(\sin x+c)=\cos x $

- $\sin x+c$ is anti derivative of  $\cos x$

- $c$ - arbitrary constant, $c \in \mathbb{R}$

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<footer style = "font-size: 18px">First Fundamental theorem of calculus $\rightarrow$ Antiderivative $\rightarrow$ <span style = "color:blue"> Antiderivative </span> $\rightarrow$ Antiderivative $\rightarrow$ Integration </footer>

### <Success style="font-size:25px"> Indefinite Integral L-1 </Success>
### <primary style="font-size:24px"> Examples </primary>
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- $ f(x)=e^{4 x} $
- $ \frac{d}{d x}\left(\frac{e^{4 x}}{4}\right)=e^{4 x} $   
- $ \Rightarrow$ Anti derivative of $e^{4 x} = \frac{e^{4 x}}{4}+ c.$
  
- 3.

- $ f(x)=\sin 2 x-4 e^{3 x} $   
- $ \frac{d}{d x}\left(-\frac{1}{2} \cos 2 x-4 \cdot \frac{e^{3 x}}{3}\right)=\sin 2 x-4 e^{3 x} $    
- $ \Rightarrow$ Anti derivative of 
- $\sin 2 x-4 e^{3 x} =  -\frac{1}{2} \cos 2 x-\frac{4}{3} e^{3 x}+c $

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<footer style = "font-size: 18px">Examples $\rightarrow$ Examples $\rightarrow$ <span style = "color:blue"> Examples </span> $\rightarrow$ Remark $\rightarrow$ Examples </footer>

### <Success style="font-size:25px"> Indefinite Integral L-1 </Success>
### <primary style="font-size:24px"> Examples </primary>
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- $ \frac{d}{d x}\left(\sin ^{-1} x\right)=\frac{1}{\sqrt{1-x^2}} $

- $ \frac{d}{d x}\left(\cos ^{-1} x\right)=-\frac{1}{\sqrt{1-x^2}} $

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

- $ \sin ^{-1} x+\cos ^{-1} x=\text { constant } $

- $ \sin ^{-1} x+\cos ^{-1} x=\pi / 2$

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<footer style = "font-size: 18px">Examples $\rightarrow$ Remark $\rightarrow$ <span style = "color:blue"> Examples </span> $\rightarrow$ Tangent at a point $\rightarrow$ Remark </footer>

### <Success style="font-size:25px"> Indefinite Integral L-1 </Success>
### <primary style="font-size:24px"> Tangent at a point </primary>
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- Let $f(x)=e^x$ and now we see tangent at the point $P_ 0, P_ 1, P_ 2..$   and the points $Q_ 0,Q_ 1,Q_2 ..$

- ${P_0} = (0,0)$

- $ \frac{d y}{d x}|_ {P_ 0} = \frac{d}{d x} \(e^x {-1})|_ {P_ 0}=  {e^x|}_ {P_0}=1,$

- ${Q_0} = (1,e-1)$

- $\frac{d y}{d x}|_ {Q_ 0}=\frac{d}{d x}(e^x-1)|_ {Q_ 0}  = e^xt|_ {Q_0}=e^1=e $

- $\frac{d y}{d x}|_{Q_1}=e$

- ${P_1} = (0,1)$

- $ \frac{d y}{d x}|_ {P_ 1} =\frac{d}{d x} \(e^x)|_ {P_ 1}= {e^x|}_ {P_ 1}=1$

- ${Q_ 1} = (1,e), ~ \frac{d y}{d x}|_ {Q_ 1} =\frac{d}{d x} \(e^x)|_ {Q _ 1} ={e^x|}_{Q _1}= e$

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<footer style = "font-size: 18px">Remark $\rightarrow$ Examples $\rightarrow$ <span style = "color:blue"> Tangent at a point </span> $\rightarrow$ Remark $\rightarrow$ Thank you </footer>