Mathematics

Evaluate the integral
$$\displaystyle \int_{0}^{a}f(x)dx+\int_{0}^{a}f(2a-x)dx$$ 


ANSWER

$$\displaystyle \int_{0}^{2a}f(x)dx$$


SOLUTION
$$I=\displaystyle\int_{0}^{a}f(x)dx+\int_{0}^{a}f(2a-x)dx$$

Using the property $$\displaystyle\int_{a}^{b}f(x)dx=\int_{a}^{b}f(a+b-x)dx$$ in the second part:-

$$\implies I=\displaystyle\int_{0}^{a}f(x)dx+\int_{0}^{a}f\left(2a-(a-x)\right)dx$$

$$\implies I=\displaystyle\int_{0}^{a}f(x)dx+\int_{0}^{a}f(a+x)dx$$

$$\implies I=\displaystyle\int_{0}^{a}f(x)dx+\int_{a}^{2a}f(x)dx$$

$$=\displaystyle\int_{0}^{2a}f(x)dx$$

Hence, answer is option-(B).
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Single Correct Medium Published on 17th 09, 2020
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