$\int \frac{1}{4-9x} \mathrm{d}x$ $\int \frac{1}{4-9x^2} \mathrm{d}x$ \(\int \frac{x}{4-9x^2} \m

Question

$\int \frac{1}{4-9x} \mathrm{d}x$ $\int \frac{1}{4-9x^2} \mathrm{d}x$ $\int \frac{x}{4-9x^2} \mathrm{d}x$ $\int \frac{x+3}{4-9x^2} \mathrm{d}x$ $\int \frac{x^2}{4-9x^2} \mathrm{d}x$ $\int \frac{x}{\sqrt{4-9x^2}} \mathrm{d}x$ Find $\int \frac{x+2}{\sqrt{1-8x-4x^2}} \mathrm{d}x$. Find $\int \frac{x}{x^2+4x+7} \mathrm{d}x$.

Accepted Answer

(i) $-\dfrac19\ln|4-9x|+C$ (ii) $\dfrac1{12}\ln\left|\dfrac{2+3x}{2-3x}\right|+C$ (iii) $-\dfrac1{18}\ln|4-9x^2|+C$ (iv) $-\dfrac1{18}\ln|4-9x^2|+\dfrac14\ln\left|\dfrac{2+3x}{2-3x}\right|+C$ (v) $-\dfrac x9+\dfrac1{27}\ln\left|\dfrac{2+3x}{2-3x}\right|+C$ (vi) $-\dfrac19\sqrt{4-9x^2}+C$ (vii) $-\dfrac14\sqrt{1-8x-4x^2}+\dfrac12\sin^{-1}\left(\dfrac{2(x+1)}{\sqrt5}\right)+C$ (viii) $\dfrac12\ln(x^2+4x+7)-\dfrac2{\sqrt3}\tan^{-1}\left(\dfrac{x+2}{\sqrt3}\right)+C$

Techniques of Integration - Integration by Splitting the Numerator

Techniques of Integration - Integration by Splitting the Numerator

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