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Question Number 59012 by cesar.marval.larez@gmail.com last updated on 03/May/19

Answered by tanmay last updated on 03/May/19

$$182)\int \frac{2x-7}{x^2+9}dx$$$$\int \frac{d(x^2+9)}{x^2+9}-7\int \frac{dx}{x^2+3^2}$$$$ln(x^2+9)-7\times \frac{1}{3}{tan}^{-1}\left(\frac{x}{3}\right)+c$$

Answered by tanmay last updated on 03/May/19

$$183)\int \frac{x+3}{\sqrt{1-x^2}}dx$$$$\int \frac{x}{\sqrt{1-x^2}}dx+3\int \frac{dx}{\sqrt{1-x^2}}$$$$x=sinadx=cosada$$$$\int \frac{sina}{cosa}\times cosada+3\int \frac{cosada}{cosa}$$$$\int sinada+3\int da$$$$=-cosa+3a+c$$$$=-\sqrt{1-x^2}+3{sin}^{-1}\left(x\right)+c$$$$$$$$$$

Answered by tanmay last updated on 03/May/19

$$\int \frac{4x+5}{4x^2+1}dx$$$$\frac{1}{2}\int \frac{d(4x^2+1)}{4x^2+1}+\int \frac{5dx}{4x^2+1}$$$$\frac{1}{2}\int \frac{d(4x^2+1)}{4x^2+1}+\frac{5}{4}\int \frac{dx}{x^2+{\left(\frac{1}{2}\right)}^2}$$$$\frac{1}{2}ln(4x^2+1)+\frac{5}{4}\times \frac{1}{\frac{1}{2}}{tan}^{-1}\left(\frac{x}{\frac{1}{2}}\right)+c$$$$\frac{1}{2}ln(4x^2+1)+\frac{5}{2}{tan}^{-1}\left(2x\right)+c$$

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