Advertisements
Advertisements
प्रश्न
\[\int\limits_0^1 \left| \sin 2\pi x \right| dx\]
Advertisements
उत्तर
We have,
\[\left| \sin2\pi x \right| = \begin{cases}\left( \sin2\pi x \right),& 0 \leq x \leq \frac{1}{2}\\ - \left( \sin2\pi x \right),& \frac{1}{2} \leq x \leq 1\end{cases}\]
\[ \therefore \int_0^1 \left| \sin2\pi x \right| d x = \int_0^\frac{1}{2} \sin2\pi x dx + \int_\frac{1}{2}^1 - \sin2\pi x dx\]
\[ = \left[ \frac{- \cos2\pi x}{2\pi} \right]_0^\frac{1}{2} + \left[ \frac{\cos2\pi x}{2\pi} \right]_\frac{1}{2}^1 \]
\[ = \frac{1}{2\pi} + \frac{1}{2\pi} + \frac{1}{2\pi} + \frac{1}{2\pi}\]
\[ = \frac{2}{\pi}\]
APPEARS IN
संबंधित प्रश्न
\[\int\limits_1^4 f\left( x \right) dx, where f\left( x \right) = \begin{cases}7x + 3 & , & \text{if }1 \leq x \leq 3 \\ 8x & , & \text{if }3 \leq x \leq 4\end{cases}\]
If \[f\left( x \right) = \int_0^x t\sin tdt\], the write the value of \[f'\left( x \right)\]
The value of \[\int\limits_0^{2\pi} \sqrt{1 + \sin\frac{x}{2}}dx\] is
If \[I_{10} = \int\limits_0^{\pi/2} x^{10} \sin x\ dx,\] then the value of I10 + 90I8 is
Evaluate: \[\int\limits_{- \pi/2}^{\pi/2} \frac{\cos x}{1 + e^x}dx\] .
\[\int\limits_0^{\pi/3} \frac{\cos x}{3 + 4 \sin x} dx\]
\[\int\limits_0^{\pi/4} \sin 2x \sin 3x dx\]
\[\int\limits_0^{\pi/4} \cos^4 x \sin^3 x dx\]
\[\int\limits_{- \pi/2}^{\pi/2} \sin^9 x dx\]
\[\int\limits_0^\pi x \sin x \cos^4 x dx\]
Using second fundamental theorem, evaluate the following:
`int_1^2 (x "d"x)/(x^2 + 1)`
Using second fundamental theorem, evaluate the following:
`int_1^"e" ("d"x)/(x(1 + logx)^3`
Evaluate the following:
`int_(-1)^1 "f"(x) "d"x` where f(x) = `{{:(x",", x ≥ 0),(-x",", x < 0):}`
Evaluate the following integrals as the limit of the sum:
`int_1^3 (2x + 3) "d"x`
Choose the correct alternative:
`int_0^oo "e"^(-2x) "d"x` is
Choose the correct alternative:
If n > 0, then Γ(n) is
Evaluate `int (x^2 + x)/(x^4 - 9) "d"x`
`int x^3/(x + 1)` is equal to ______.
Find: `int logx/(1 + log x)^2 dx`
