Mathematics

Evaluate the following:
$$\displaystyle\int \dfrac { 2 x + 7 } { ( x - 4 ) ^ { 2 } } d x$$


SOLUTION
$$I=\displaystyle\int\dfrac{2x+7}{(x-4)^2}dx$$

$$I=\displaystyle\int\dfrac{2x-8+15}{(x-4)^2}dx=\displaystyle\int\dfrac{2x-8}{(x-4)^2}dx+\displaystyle\int\dfrac{15}{(x-4)^2}dx$$

$$I=\displaystyle\int\dfrac{2(x-4)}{(x-4)^2}dx+15\displaystyle\int\dfrac{1}{(x-4)^2}dx$$

Let $$x-4=t\Rightarrow dx=dt$$

$$I=\displaystyle2\int \dfrac{dx}{t}+15\int \dfrac{1}{t^2}dt$$

$$I=2\log(t)+15\dfrac{t^{-1}}{-1}$$

$$I=2log(t)-\dfrac{15}{t}$$

$$I=2log (x-4)-\dfrac{15}{(x-4)}+c$$.
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Subjective Medium Published on 17th 09, 2020
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