Integrate the function -x2-4 x-1 d x

Let I =∫√(x^2 +4x+1)dx =∫√(x^2 +4x+1 2^2+2^2)dx =∫√((x+2)^2 (√(3))^2)dx [ ∵∫√(x^2 a^2)dx =x/2√(x^2 a^2) a^2/2log |x+√(x^2 a^2)| ] ⇒ I =x+2/2√(x^2 +4x+1) 3/2 ...

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Standard XII

Mathematics

Integration of Irrational Algebraic Fractions - 2

Integrate the...

Question

Integrate the function.
$\int \sqrt{x^{2} + 4 x + 1} d x .$

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Solution

Let $I = \int \sqrt{x^{2} + 4 x + 1} d x = \int \sqrt{x^{2} + 4 x + 1 - 2^{2} + 2^{2}} d x$
$= \int \sqrt{(x + 2)^{2} - (\sqrt{3})^{2}} d x [∵ \int \sqrt{x^{2} - a^{2}} d x = \frac{x}{2} \sqrt{x^{2} - a^{2}} - \frac{a^{2}}{2} l o g | x + \sqrt{x^{2} - a^{2}} |] \Rightarrow I = \frac{x + 2}{2} \sqrt{x^{2} + 4 x + 1} - \frac{3}{2} l o g | (x + 2) + \sqrt{x^{2} + 4 x + 1} | + C$

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Integrate the function. ∫√x2+4x+1dx.

Let I=∫√x2+4x+1dx=∫√x2+4x+1−22+22dx =∫√(x+2)2−(√3)2dx[∵∫√x2−a2dx=x2√x2−a2−a22log|x+√x2−a2|]⇒I=x+22√x2+4x+1−32log|(x+2)+√x2+4x+1|+C

Integrate the function.
$\int \sqrt{x^{2} + 4 x + 1} d x .$