Integrate the function -x2-4 x-6 d x

Let I =∫√(x^2+4x+6)dx =∫√(x^2 +4x+2^2 +6 4)dx =∫√((x+2)^2+(√(2))^2)dx [ ∵∫√(x^2 +a^2)dx =x/2√(x^2+a^2)+a^2/2log |x+√(x^2+a^2)| ] =x+2/2√(x^2+4x+6)+2/2log |(x ...

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Byju's Answer

Standard XII

Mathematics

Integration of Irrational Algebraic Fractions - 2

Integrate the...

Question

Integrate the function.
$\int \sqrt{x^{2} + 4 x + 6} d x .$

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Solution

Let $I = \int \sqrt{x^{2} + 4 x + 6} d x = \int \sqrt{x^{2} + 4 x + 2^{2} + 6 - 4} d x$
$= \int \sqrt{(x + 2)^{2} + (\sqrt{2})^{2}} d x [∵ \int \sqrt{x^{2} + a^{2}} d x = \frac{x}{2} \sqrt{x^{2} + a^{2}} + \frac{a^{2}}{2} l o g | x + \sqrt{x^{2} + a^{2}} |] = \frac{x + 2}{2} \sqrt{x^{2} + 4 x + 6} + \frac{2}{2} l o g | (x + 2) + \sqrt{x^{2} + 4 x + 6} | + C \Rightarrow I = \frac{x + 2}{2} \sqrt{x^{2} + 4 x + 6} + l o g | (x + 2) + \sqrt{x^{2} + 4 x + 6} | + C$

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Integrate the function. ∫√x2+4x+6dx.

Let I=∫√x2+4x+6dx=∫√x2+4x+22+6−4dx =∫√(x+2)2+(√2)2dx[∵∫√x2+a2dx=x2√x2+a2+a22log|x+√x2+a2|]=x+22√x2+4x+6+22log|(x+2)+√x2+4x+6|+C⇒I=x+22√x2+4x+6+log|(x+2)+√x2+4x+6|+C

Integrate the function.
$\int \sqrt{x^{2} + 4 x + 6} d x .$