Advertisements
Advertisements
प्रश्न
\[\int\limits_0^{\pi/2} \frac{\sin^2 x}{\left( 1 + \cos x \right)^2} dx\]
Advertisements
उत्तर
\[\int_0^\frac{\pi}{2} \frac{\sin^2 x}{\left( 1 + \cos x \right)^2} d x\]
\[ = \int_0^\frac{\pi}{2} \frac{1 - \cos^2 x}{\left( 1 + \cos x \right)^2} d x\]
\[ = \int_0^\frac{\pi}{2} \frac{\left( 1 + \cos x \right)\left( 1 - \cos x \right)}{\left( 1 + \cos x \right)^2} d x\]
\[ = \int_0^\frac{\pi}{2} \frac{1 - \cos x}{1 + \cos x} d x\]
\[ = \int_0^\frac{\pi}{2} \frac{1 - \cos x - 1 + 1}{\left( 1 + \cos x \right)} d x\]
\[ = \int_0^\frac{\pi}{2} \frac{2 - \left( 1 + \cos x \right)}{\left( 1 + \cos x \right)} d x\]
\[ = \int_0^\frac{\pi}{2} \frac{2}{1 + \cos x}dx - \int_0^\frac{\pi}{2} dx\]
\[ = \int_0^\frac{\pi}{2} \frac{2\left( 1 - \cos x \right)}{\left( 1 + \cos x \right)\left( 1 - \cos x \right)}dx - \int_0^\frac{\pi}{2} dx\]
\[ = 2 \int_0^\frac{\pi}{2} \frac{1 - \cos x}{\sin^2 x}dx - \left[ x \right]_0^\frac{\pi}{2} \]
\[ = 2 \int_0^\frac{\pi}{2} \left( \ cosec^2 x - \ cosec x\ cotx \right) dx - \left[ x \right]_0^\frac{\pi}{2} \]
\[ = 2 \left[ - cotx + \ cosec x \right]_0^\frac{\pi}{2} - \left[ x \right]_0^\frac{\pi}{2} \]
\[ = 2 - \frac{\pi}{2}\]
APPEARS IN
संबंधित प्रश्न
Evaluate the following integral:
If f is an integrable function, show that
\[\int\limits_{- a}^a f\left( x^2 \right) dx = 2 \int\limits_0^a f\left( x^2 \right) dx\]
If \[f\left( x \right) = \int_0^x t\sin tdt\], the write the value of \[f'\left( x \right)\]
Write the coefficient a, b, c of which the value of the integral
\[\int\limits_0^1 \left\{ x \right\} dx,\] where {x} denotes the fractional part of x.
The value of \[\int\limits_{- \pi}^\pi \sin^3 x \cos^2 x\ dx\] is
If \[I_{10} = \int\limits_0^{\pi/2} x^{10} \sin x\ dx,\] then the value of I10 + 90I8 is
The value of \[\int\limits_0^{\pi/2} \log\left( \frac{4 + 3 \sin x}{4 + 3 \cos x} \right) dx\] is
Evaluate : \[\int\limits_0^\pi/4 \frac{\sin x + \cos x}{16 + 9 \sin 2x}dx\] .
\[\int\limits_1^5 \frac{x}{\sqrt{2x - 1}} dx\]
\[\int\limits_0^{\pi/2} \frac{\sin x}{\sqrt{1 + \cos x}} dx\]
\[\int\limits_0^{\pi/2} \left| \sin x - \cos x \right| dx\]
\[\int\limits_0^a \frac{\sqrt{x}}{\sqrt{x} + \sqrt{a - x}} dx\]
\[\int\limits_{\pi/6}^{\pi/2} \frac{\ cosec x \cot x}{1 + {cosec}^2 x} dx\]
Evaluate the following:
f(x) = `{{:("c"x",", 0 < x < 1),(0",", "otherwise"):}` Find 'c" if `int_0^1 "f"(x) "d"x` = 2
Evaluate the following using properties of definite integral:
`int_(- pi/2)^(pi/2) sin^2theta "d"theta`
Evaluate the following integrals as the limit of the sum:
`int_0^1 (x + 4) "d"x`
Evaluate the following integrals as the limit of the sum:
`int_1^3 (2x + 3) "d"x`
Choose the correct alternative:
`Γ(3/2)`
`int x^3/(x + 1)` is equal to ______.
