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Fundamental Laws of Logarithms

∫ 2 x+5 x 2+2...

Question

$\int \frac{2 x + 5}{\sqrt{x^{2} + 2 x + 5}} d x$

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Solution

$Let I = \int \frac{(2 x + 5) d x}{\sqrt{x^{2} + 2 x + 5}} Consider, 2 x + 5 = A \frac{d}{d x} (x^{2} + 2 x + 5) + B \Rightarrow 2 x + 5 = A (2 x + 2) + B \Rightarrow 2 x + 5 = (2 A) x + 2 A + B Equating Coefficients of like terms 2 A = 2 \Rightarrow A = 1 And 2 A + B = 5 \Rightarrow B = 3 ∴ I = \int (\frac{2 x + 2 + 3}{\sqrt{x^{2} + 2 x + 5}}) d x = \int \frac{(2 x + 2) d x}{\sqrt{x^{2} + 2 x + 5}} + 3 \int \frac{d x}{\sqrt{x^{2} + 2 x + 5}} let x^{2} + 2 x + 5 = t \Rightarrow (2 x + 2) d x = d t Then, I = \int \frac{d t}{\sqrt{t}} + 3 \int \frac{d x}{\sqrt{x^{2} + 2 x + 1 + 4}} = \int t^{- \frac{1}{2}} d t + 3 \int \frac{d x}{\sqrt{{(x + 1)}^{2} + 2^{2}}} = \frac{t^{- \frac{1}{2} + 1}}{- \frac{1}{2} + 1} + 3 \log |x + 1 + \sqrt{{(x + 1)}^{2} + 4}| + C = 2 \sqrt{t} + 3 \log |x + 1 + \sqrt{x^{2} + 2 x + 5}| + C = 2 \sqrt{x^{2} + 2 x + 5} + 3 \log |x + 1 + \sqrt{x^{2} + 2 x + 5}| + C$

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