Integrate the following functions- -1-x2-2 x-2 d x

Let I =∫1/√(x^2+2x+2)dx =∫1/√((x^2+2x+1)+1)dx =∫1/√((x+1)^2 +1)dx Let x+1 =t ⇒ dx =dt ∴ I =∫1/√(t^2+1)dt =log |t+√(t^2 +1)|+C [ ∵∫dx/√(x^2+a^2)=log |x+√(x^2+a^ ...

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Standard XII

Mathematics

Theorems on Integration

Integrate the...

Question

Integrate the following functions.
$\int \frac{1}{\sqrt{x^{2} + 2 x + 2}} d x .$

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Solution

Let $I = \int \frac{1}{\sqrt{x^{2} + 2 x + 2}} d x = \int \frac{1}{\sqrt{(x^{2} + 2 x + 1) + 1}} d x$
$= \int \frac{1}{\sqrt{(x + 1)^{2} + 1}} d x$
Let $x + 1 = t \Rightarrow d x = d t$
$∴ I = \int \frac{1}{\sqrt{t^{2} + 1}} d t = l o g | t + \sqrt{t^{2} + 1} | + C [∵ \int \frac{d x}{\sqrt{x^{2} + a^{2}}} = l o g | x + \sqrt{x^{2} + a^{2}} |] = l o g | (x + 1) + \sqrt{(x + 1)^{2} + 1} | + C = l o g | (x + 1) + \sqrt{x^{2} + 2 x + 2} | + C (∵ t = x + 1)$

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Integrate the following functions. ∫1√x2+2x+2dx.

Let I=∫1√x2+2x+2dx=∫1√(x2+2x+1)+1dx =∫1√(x+1)2+1dx Let x+1=t⇒dx=dt ∴I=∫1√t2+1dt=log|t+√t2+1|+C[∵∫dx√x2+a2=log|x+√x2+a2|]=log|(x+1)+√(x+1)2+1|+C=log|(x+1)+√x2+2x+2|+C(∵t=x+1)

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