Mathematics

Evaluate:$$\displaystyle\int{\dfrac{x\,dx}{a+bx}}$$


SOLUTION
$$\displaystyle\int{\dfrac{x\,dx}{a+bx}}$$
$$=\displaystyle\int{\dfrac{\left(a+bx-a\right)\,dx}{a+bx}}$$
$$=\displaystyle\int{\dfrac{\left(a+bx\right)\,dx}{a+bx}}-\displaystyle\int{\dfrac{a\,dx}{a+bx}}$$
$$=\displaystyle\int{dx}-\dfrac{a}{b}\displaystyle\int{\dfrac{b\,dx}{a+bx}}$$
Let $$t=a+bx\Rightarrow\,dt=b\,dx$$
$$=\displaystyle\int{dx}-\dfrac{a}{b}\displaystyle\int{\dfrac{dt}{t}}$$
$$=\displaystyle\int{dx}-\dfrac{a}{b}\ln{\left|t\right|}+c$$
$$=x-\dfrac{a}{b}\ln{\left|\left(a+bx\right)\right|}+c$$
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Subjective Medium Published on 17th 09, 2020
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