P=√(x^2+2x+5)dx P=∫√(x^2+2x+1+4dx)=∫√((x+1)^2+(2)^2)dx We know ∫√(x^2+a^2)=12√(x^2+a^2)+a^22ln|x+√(x^2+a^2)| P=∫√((x+1)^2+(2)^2)dx P=12 ...

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Integration of Irrational Algebraic Fractions - 2

Solve: ∫√x2...

Question

Solve: $\int \sqrt{x^{2} + 2 x + 5}$ dx;

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Solution

$P = \sqrt{x^{2} + 2 x + 5} d x$
$P = \int \sqrt{x^{2} + 2 x + 1 + 4 d x} = \int \sqrt{(x + 1)^{2} + (2)^{2}} d x$
We know $\int \sqrt{x^{2} + a^{2}} = \frac{1}{2} \sqrt{x^{2} + a^{2}} + \frac{a^{2}}{2} l n | x + \sqrt{x^{2} + a^{2}} |$
$P = \int \sqrt{(x + 1)^{2} + (2)^{2}} d x$
$P = \frac{1}{2} \sqrt{x^{2} + 2 x + 5} + 2 l n | x + 1 + \sqrt{x^{2} + 2 x + 5} |$ .

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