Question

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Answer 1

EXPLANATION.

$\sf \implies \displaystyle \int \sqrt{3 - 2x - x^{2} } dx$

As we know that,

We can write equation as,

$\sf \implies \displaystyle \int \sqrt{3 - (2x + x^{2} )} dx$

Add and subtract 1 in the equation, we get

$\sf \implies \displaystyle \int \sqrt{3 - (x^{2} + 2x + 1 - 1) } dx$

$\sf \implies \displaystyle \int \sqrt{4 - (x^{2} + 2x + 1)} dx$

$\sf \implies \displaystyle \int \sqrt{4 - (x + 1)^{2} } dx$

$\sf \implies \displaystyle \int \sqrt{(2)^{2} - (x + 1)^{2} } dx$

As we know that,

Formula of :

$\sf \implies \displaystyle \int \sqrt{a^{2} - x^{2} } dx = \dfrac{x}{2} \sqrt{a^{2} - x^{2} } + \dfrac{a^{2} }{2} sin^{-1} \bigg( \dfrac{x}{a} \bigg) + C.$

Using this formula in the equation, we get.

$\sf \implies \displaystyle \dfrac{(x + 1)}{2} \sqrt{(2)^{2} - (x + 1)^{2} } + \dfrac{(2)^{2} }{2} sin^{-1} \bigg( \dfrac{(x + 1)}{2} \bigg) + C.$

$\sf \implies \displaystyle \dfrac{(x + 1)}{2} \sqrt{4 - (x^{2} + 1 + 2x)} + 2sin^{-1} \bigg( \dfrac{(x + 1)}{2} \bigg) + C.$

$\sf \implies \displaystyle \dfrac{(x + 1)}{2} \sqrt{3 - 2x - x^{2} } + 2sin^{-1} \bigg( \dfrac{(x + 1)}{2} \bigg) + C.$

Answer 2

Answer:

G I V E N :

$\:\:\:\:\:\:\:\:\large{\bold{\int \sqrt{3 \: - \: 2x \: - \: x^2}\:\: dx\: =}}$

• Add and substrate 1 ,

$\large{\bold{\Rightarrow \int \sqrt{3 - \: 2x \: - x^2\: - 1\: + 1}\:dx\: =}}$

$\:\large{\bold{\Rightarrow \int \sqrt{4 - \: 2x \: - x^2\: - 1}\:\:dx\: =}}$

$\:\large{\bold{\Rightarrow \int \sqrt{4 - \: ( \: 2x \: + x^2\: + 1\:)}\:dx\: =}}$

$\:\large{\bold{\Rightarrow \int \sqrt{2² - \: ( \: x\: + \: 1\:)^2}\:dx\: =}}$

( As we know the formula )

$\boxed{\bold{\small{\underline{\red{\int \sqrt{a²-x²}\:dx\: = \: \frac{x}{2}\sqrt{a²-x²}+\frac{1}{2}a²Sin^{-1}\frac{x}{a}+C}}}}}$

So ,

$\bold{\Rightarrow \int \sqrt{2² - \: ( \: x\: + \: 1\:)^2}\:dx\: =}$

$\small{\bold{\Rightarrow \frac{(x+1)}{2}\sqrt{2²-(x+1)²}+\frac{1}{2}2²Sin^{-1}\frac{(x+1)}{2}+C }}$

$\bold{\Rightarrow \small{\bold{\frac{(x+1)}{2}\sqrt{2²-(x+1)²}+2Sin^{-1}\frac{(x+1)}{2}+C }}}$

$\bold{\Rightarrow \small{\bold{\frac{(x+1)}{2}\sqrt{3-2x+x^2}+2Sin^{-1}\frac{(x+1)}{2}+C }}}$

$\bold{\boxed{\begin{aligned} \red{\underline{your \: Answer} \:} \\ \\ \int \sqrt{3 \: - \: 2x \: - \: x^2}\:\: dx\: = \: \frac{(x+1)}{2}\sqrt{3-2x+x^2}+2Sin^{-1}\frac{(x+1)}{2}+C \end{aligned} }}$