Advertisements
Advertisements
प्रश्न
\[\int\limits_0^a \frac{\sqrt{x}}{\sqrt{x} + \sqrt{a - x}} dx\]
Advertisements
उत्तर
\[Let, I = \int_0^a \frac{\sqrt{x}}{\sqrt{x} + \sqrt{a - x}} d x ...............(1)\]
\[ = \int_0^a \frac{\sqrt{a - x}}{\sqrt{a - x} + \sqrt{a - a + x}} d x\]
\[ = \int_0^a \frac{\sqrt{a - x}}{\sqrt{a - x} + \sqrt{x}} d x\]
\[ \Rightarrow I = \int_0^a \frac{\sqrt{a - x}}{\sqrt{x} + \sqrt{a - x}} d x.................(2)\]
Adding (1) and (2)
\[2I = \int_0^a \left[ \frac{\sqrt{x}}{\sqrt{x} + \sqrt{a - x}} + \frac{\sqrt{a - x}}{\sqrt{x} + \sqrt{a - x}} \right] d x\]
\[ = \int_0^a dx\]
\[ = \left[ x \right]_0^a \]
\[ = a\]
\[\text{Hence, }I = \frac{a}{2}\]
APPEARS IN
संबंधित प्रश्न
\[\int\limits_{\pi/4}^{\pi/2} \cot x\ dx\]
Write the coefficient a, b, c of which the value of the integral
The value of \[\int\limits_0^{\pi/2} \log\left( \frac{4 + 3 \sin x}{4 + 3 \cos x} \right) dx\] is
\[\int\limits_1^5 \frac{x}{\sqrt{2x - 1}} dx\]
\[\int\limits_0^{\pi/3} \frac{\cos x}{3 + 4 \sin x} dx\]
\[\int\limits_0^{\pi/4} \cos^4 x \sin^3 x dx\]
Evaluate the following integrals :-
\[\int_2^4 \frac{x^2 + x}{\sqrt{2x + 1}}dx\]
\[\int\limits_0^{\pi/4} e^x \sin x dx\]
\[\int\limits_{- a}^a \frac{x e^{x^2}}{1 + x^2} dx\]
\[\int\limits_0^\pi \frac{dx}{6 - \cos x}dx\]
Using second fundamental theorem, evaluate the following:
`int_0^1 "e"^(2x) "d"x`
Evaluate the following using properties of definite integral:
`int_(- pi/2)^(pi/2) sin^2theta "d"theta`
Evaluate the following:
`Γ (9/2)`
Find `int x^2/(x^4 + 3x^2 + 2) "d"x`
Find `int sqrt(10 - 4x + 4x^2) "d"x`
Evaluate `int (x^2"d"x)/(x^4 + x^2 - 2)`
If `int (3"e"^x - 5"e"^-x)/(4"e"6x + 5"e"^-x)"d"x` = ax + b log |4ex + 5e –x| + C, then ______.
`int x^9/(4x^2 + 1)^6 "d"x` is equal to ______.
Which notation represents a definite integral?
Which integral has lower and upper limits?
Evaluate \[\int_{1}^{3}(2x+1)\,dx\].
