₀^{a}{{x}}{{x}+{a-x}}dx - NCERT Class XII Mathematics Chapter

NCERT Class XII Mathematics Chapter - - Solutions

Exercise - 7.11
By using the properties of definite integrals, evaluate the integrals in Exercises 208 to 225.

Question : 223 of 271

Marks: +1, -0

\int\limits_{0}^{a}\frac{\sqrt{x}}{\sqrt{x}+\sqrt{a-x}}dx

Solution:

Let

I=\int\limits_{0}^{a}\frac{\sqrt{x}}{\sqrt{x}+\sqrt{a-x}}dx

...(i)

I=\int\limits_{0}^{a}\frac{\sqrt{a-x}}{\sqrt{a-x}+\sqrt{a-(a-x)}}dx

=\int\limits_{0}^{a}\frac{\sqrt{a-x}}{\sqrt{a-x}+\sqrt{x}}dx

...(ii)

Adding (i) and (ii), we get

2I=\int\limits_{0}^{a}\frac{\sqrt{x}+\sqrt{a-x}}{\sqrt{x}+\sqrt{a-x}}dx

=\int\limits_{0}^{a}1\cdot dx=[x]_{0}^{a}=a-0=a

\therefore I=\frac{a}{2}

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