Advertisements
Advertisements
प्रश्न
Advertisements
उत्तर
\[Let I = \int_{- \frac{\pi}{2}}^\frac{\pi}{2} \sin^3 x\ d x\]
\[ = \int_{- \frac{\pi}{2}}^\frac{\pi}{2} \sin x \sin^2 x\ dx\]
\[ = \int_{- \frac{\pi}{2}}^\frac{\pi}{2} \sin x\left( 1 - \cos^2 x \right) dx\]
\[Let\ \cos x = t, then - \sin x\ dx = dt, \]
\[When\, x \to - \frac{\pi}{2} ; t \to 0\ and\ x \to \frac{\pi}{2} ; t \to 0\]
\[I = \int_0^0 \left( - 1 + t^2 \right) dt\]
\[\]
\[ = 0\]
APPEARS IN
संबंधित प्रश्न
Evaluate the following definite integrals:
\[\int\limits_1^4 f\left( x \right) dx, where f\left( x \right) = \begin{cases}7x + 3 & , & \text{if }1 \leq x \leq 3 \\ 8x & , & \text{if }3 \leq x \leq 4\end{cases}\]
If `f` is an integrable function such that f(2a − x) = f(x), then prove that
If f(2a − x) = −f(x), prove that
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
\[\int_0^\frac{\pi^2}{4} \frac{\sin\sqrt{x}}{\sqrt{x}} dx\] equals
Evaluate: \[\int\limits_{- \pi/2}^{\pi/2} \frac{\cos x}{1 + e^x}dx\] .
Evaluate : \[\int e^{2x} \cdot \sin \left( 3x + 1 \right) dx\] .
\[\int\limits_0^1 \cos^{- 1} x dx\]
\[\int\limits_0^1 \cos^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right) dx\]
\[\int\limits_0^{\pi/2} \frac{\sin x}{\sqrt{1 + \cos x}} dx\]
\[\int\limits_0^{\pi/2} \frac{\cos x}{1 + \sin^2 x} dx\]
\[\int\limits_0^\pi \frac{x}{a^2 \cos^2 x + b^2 \sin^2 x} dx\]
Using second fundamental theorem, evaluate the following:
`int_1^2 (x "d"x)/(x^2 + 1)`
Evaluate the following:
`int_0^2 "f"(x) "d"x` where f(x) = `{{:(3 - 2x - x^2",", x ≤ 1),(x^2 + 2x - 3",", 1 < x ≤ 2):}`
Evaluate the following using properties of definite integral:
`int_0^(i/2) (sin^7x)/(sin^7x + cos^7x) "d"x`
Evaluate the following integrals as the limit of the sum:
`int_0^1 (x + 4) "d"x`
Choose the correct alternative:
`Γ(3/2)`
Verify the following:
`int (2x + 3)/(x^2 + 3x) "d"x = log|x^2 + 3x| + "C"`
Evaluate: `int_(-1)^2 |x^3 - 3x^2 + 2x|dx`
