Advertisements
Advertisements
प्रश्न
\[\int\limits_0^\pi x \sin x \cos^4 x dx\]
Advertisements
उत्तर
\[Let, I = \int_0^\pi x \sin x \cos^4 x d x ............(1)\]
\[ = \int_0^\pi \left( \pi - x \right) \sin\left( \pi - x \right) \cos^4 \left( \pi - x \right) d x \]
\[ = \int_0^\pi \left( \pi - x \right) \sin x \cos^4 x d x ..............(2)\]
Adding (1) and (2)
\[2I = \int_0^\pi \left[ x \sin x \cos^4 x + \left( \pi - x \right) \sin x \cos^4 x \right] d x \]
\[ = \int_0^\pi \left( x + \pi - x \right) \sin x \cos^4 x d x \]
\[ = \pi \int_0^\pi \sin x \cos^4 x d x \]
\[ = \pi \left[ \frac{- \cos^5 x}{5} \right]_0^\pi \]
\[ = \pi\left[ \frac{1}{5} + \frac{1}{5} \right]\]
\[ = \frac{2\pi}{5}\]
\[Hence, I = \frac{\pi}{5}\]
APPEARS IN
संबंधित प्रश्न
\[\int\limits_{\pi/4}^{\pi/2} \cot x\ dx\]
\[\int\limits_1^4 f\left( x \right) dx, where f\left( x \right) = \begin{cases}7x + 3 & , & \text{if }1 \leq x \leq 3 \\ 8x & , & \text{if }3 \leq x \leq 4\end{cases}\]
Evaluate each of the following integral:
If \[\int\limits_0^a 3 x^2 dx = 8,\] write the value of a.
Evaluate :
The value of \[\int\limits_0^\pi \frac{x \tan x}{\sec x + \cos x} dx\] is __________ .
\[\int\limits_0^\infty \frac{1}{1 + e^x} dx\] equals
`int_0^1 sqrt((1 - "x")/(1 + "x")) "dx"`
\[\int\limits_0^1 \frac{1 - x}{1 + x} dx\]
\[\int\limits_0^1 \sqrt{\frac{1 - x}{1 + x}} dx\]
\[\int\limits_{\pi/3}^{\pi/2} \frac{\sqrt{1 + \cos x}}{\left( 1 - \cos x \right)^{5/2}} dx\]
\[\int\limits_0^a \frac{\sqrt{x}}{\sqrt{x} + \sqrt{a - x}} dx\]
\[\int\limits_2^3 \frac{\sqrt{x}}{\sqrt{5 - x} + \sqrt{x}} dx\]
\[\int\limits_{- \pi}^\pi x^{10} \sin^7 x dx\]
`int x^3/(x + 1)` is equal to ______.
Given `int "e"^"x" (("x" - 1)/("x"^2)) "dx" = "e"^"x" "f"("x") + "c"`. Then f(x) satisfying the equation is:
