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Question Number 128752 by naka3546 last updated on 10/Jan/21

$$\int \frac{dx}{2x^2+12x+9}=?$$

Answered by mr W last updated on 10/Jan/21

$$\int \frac{dx}{2x^2+12x+9}$$$$=\int \frac{dx}{2{(x+3)}^2-9}$$$$=\frac{1}{2}\int \frac{d(x+3)}{{(x+3)}^2-{\left(\frac{3}{\sqrt{2}}\right)}^2}\Rightarrow \int \frac{dx}{x^2-a^2}=\frac{1}{2a}\ln \mid \frac{x-a}{x+a}\mid +C$$$$=\frac{1}{2}\times \frac{1}{2\left(\frac{3}{\sqrt{2}}\right)}\ln \mid \frac{x+3-\frac{3}{\sqrt{2}}}{x+3+\frac{3}{\sqrt{2}}}\mid +C$$$$=\frac{\sqrt{2}}{12}\ln \mid \frac{\sqrt{2}x+3(\sqrt{2}-1)}{\sqrt{2}x+3(\sqrt{2}+1)}\mid +C$$

Commented by naka3546 last updated on 10/Jan/21

$$\mathrm{thank}\mathrm{you},sir.$$

Answered by Dwaipayan Shikari last updated on 10/Jan/21

$$\frac{1}{2}\int \frac{dx}{{(x+3)}^2-{\left(\frac{3}{\sqrt{2}}\right)}^2}=\frac{1}{6\sqrt{2}}log\left(\frac{\sqrt{2}x+3\sqrt{2}-3}{\sqrt{2}x+3\sqrt{2}+3}\right)+C$$

Commented by naka3546 last updated on 10/Jan/21

$$thankyou,sir.$$

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