Advertisements
Advertisements
प्रश्न
The value of \[\int\limits_{- \pi}^\pi \sin^3 x \cos^2 x\ dx\] is
विकल्प
- \[\frac{\pi^4}{2}\]
- \[\frac{\pi^4}{4}\]
0
none of these
Advertisements
उत्तर
0
\[\int_{- \pi}^\pi \sin^3 x \cos^2 x d x\]
\[ = \int_{- \pi}^\pi \sin x\left( 1 - \cos^2 x \right) \cos^2 x dx\]
\[Let\ \cos x = t, then - \sin x dx = dt, \]
\[When\, x = - \pi, t = - 1, x = \pi, t = - 1\]
\[\text{Therefore the integral becomes}\]
\[ \int_{- 1}^{- 1} - \left( 1 - t^2 \right) t^2 dt\]
\[ = 0\]
APPEARS IN
संबंधित प्रश्न
\[\int\limits_{\pi/4}^{\pi/2} \cot x\ dx\]
\[\int\limits_0^{( \pi )^{2/3}} \sqrt{x} \cos^2 x^{3/2} dx\]
If \[f\left( a + b - x \right) = f\left( x \right)\] , then prove that \[\int_a^b xf\left( x \right)dx = \frac{a + b}{2} \int_a^b f\left( x \right)dx\]
If \[\int\limits_0^1 \left( 3 x^2 + 2x + k \right) dx = 0,\] find the value of k.
\[\int\limits_0^1 \left\{ x \right\} dx,\] where {x} denotes the fractional part of x.
The value of the integral \[\int\limits_0^\infty \frac{x}{\left( 1 + x \right)\left( 1 + x^2 \right)} dx\]
If \[\int\limits_0^1 f\left( x \right) dx = 1, \int\limits_0^1 xf\left( x \right) dx = a, \int\limits_0^1 x^2 f\left( x \right) dx = a^2 , then \int\limits_0^1 \left( a - x \right)^2 f\left( x \right) dx\] equals
The value of the integral \[\int\limits_{- 2}^2 \left| 1 - x^2 \right| dx\] is ________ .
Evaluate: \[\int\limits_{- \pi/2}^{\pi/2} \frac{\cos x}{1 + e^x}dx\] .
\[\int\limits_0^1 \tan^{- 1} x dx\]
\[\int\limits_0^{\pi/2} \frac{\sin^2 x}{\left( 1 + \cos x \right)^2} dx\]
\[\int\limits_0^{\pi/2} \frac{\sin x}{\sqrt{1 + \cos x}} dx\]
\[\int\limits_0^{\pi/2} x^2 \cos 2x dx\]
\[\int\limits_0^\pi \frac{x}{a^2 \cos^2 x + b^2 \sin^2 x} dx\]
\[\int\limits_{- 1}^1 e^{2x} dx\]
Evaluate the following using properties of definite integral:
`int_(- pi/2)^(pi/2) sin^2theta "d"theta`
Evaluate the following:
Γ(4)
Evaluate the following integrals as the limit of the sum:
`int_0^1 (x + 4) "d"x`
Find `int x^2/(x^4 + 3x^2 + 2) "d"x`
Find `int sqrt(10 - 4x + 4x^2) "d"x`
Which integral has lower and upper limits?
