Mathematics

What is the value of $$\int_{0}^{a}\dfrac{x-a}{x+a}\ dx$$?


ANSWER

$$a-2a\log 2$$


SOLUTION
$$I= _{0}^{a}\int \dfrac{x-a}{x+a} dx$$

Let $$t =x+a \Rightarrow dt =dx$$

When $$x= 0, t=a$$

$$x=a, t= 2a$$

Substituting, $$I=  _{a}^{2a} \int \dfrac{t-2a}{t} dt$$

$$= [t-2a \log |t|]^{2a}_{a}$$ 
$$=2a-2a \log 2a- a+2a \log a$$
$$=a-2a\log 2$$

$$\therefore $$ Option $$B$$ is correct.
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Single Correct Medium Published on 17th 09, 2020
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