Advertisements
Advertisements
प्रश्न
उत्तर
We have,
\[\int\limits_0^9 f\left( x \right) dx, where f\left( x \right) \begin{cases}\sin x & , & 0 \leq x \leq \frac{\pi}{2} \\ 1 & , & \frac{\pi}{2} \leq x \leq 3 \\ e^{x - 3} & , & 3 \leq x \leq 9\end{cases}\]
\[I = \int_0^9 f\left( x \right) d x\]
\[ \Rightarrow I = \int_0^\frac{\pi}{2} f\left( x \right) d x + \int_\frac{\pi}{2}^3 f\left( x \right) d x + \int_3^9 f\left( x \right) d x ....................\left[\text{Additive property} \right]\]
\[ \Rightarrow I = \int_0^\frac{\pi}{2} \sin x d x + \int_\frac{\pi}{2}^3 1 d x + \int_3^9 e^{x - 3} d x\]
\[ \Rightarrow I = \left[ - \cos x \right]_0^\frac{\pi}{2} + \left[ x \right]_\frac{\pi}{2}^3 + \left[ e^{x - 3} \right]_3^9 \]
\[ \Rightarrow I = 0 + 1 + 3 - \frac{\pi}{2} + e^6 - e^0 \]
\[ \Rightarrow I = 3 - \frac{\pi}{2} + e^6\]
APPEARS IN
संबंधित प्रश्न
Evaluate the following integral:
Evaluate each of the following integral:
Solve each of the following integral:
Write the coefficient a, b, c of which the value of the integral
The value of \[\int\limits_0^{\pi/2} \cos x\ e^{\sin x}\ dx\] is
The derivative of \[f\left( x \right) = \int\limits_{x^2}^{x^3} \frac{1}{\log_e t} dt, \left( x > 0 \right),\] is
The value of the integral \[\int\limits_{- 2}^2 \left| 1 - x^2 \right| dx\] is ________ .
Evaluate : \[\int\limits_0^\pi/4 \frac{\sin x + \cos x}{16 + 9 \sin 2x}dx\] .
\[\int\limits_0^{\pi/3} \frac{\cos x}{3 + 4 \sin x} dx\]
\[\int\limits_0^{\pi/2} \left| \sin x - \cos x \right| dx\]
\[\int\limits_1^3 \left| x^2 - 4 \right| dx\]
\[\int\limits_0^\pi \frac{x}{1 + \cos \alpha \sin x} dx\]
Using second fundamental theorem, evaluate the following:
`int_0^1 x"e"^(x^2) "d"x`
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:
`Γ (9/2)`
Evaluate the following integrals as the limit of the sum:
`int_1^3 (2x + 3) "d"x`
Find `int x^2/(x^4 + 3x^2 + 2) "d"x`
