Question:
If $I_{m}=\int\limits_{1}^{e}(\ln x)^{m} d x$, where $m \in N$, then $I_{10}+10 I_{9}$ is equal to -
If $I_{m}=\int\limits_{1}^{e}(\ln x)^{m} d x$, where $m \in N$, then $I_{10}+10 I_{9}$ is equal to -
Updated On: Jan 30, 2025
- $e^{10}$
- $\frac{e^{10}}{10}$
- $e$
- $e^{ -1}$
Hide Solution
Verified By Collegedunia
The Correct Option is C
Solution and Explanation
$I_{10}=\int\limits_{1}^{e} 1 .(\ln x)^{10} d x=\left[(\ln x)^{10} x\right]_{1}^{e}$
$-\int\limits_{-1}^{e} 10(\ln x)^{9} \cdot \frac{1}{x} \cdot x d x$
$=e-0-10 \int\limits_{1}^{e}(\ln x)^{9} d x$
$=e-10 I_{9}+10 I_{9}=e$
$-\int\limits_{-1}^{e} 10(\ln x)^{9} \cdot \frac{1}{x} \cdot x d x$
$=e-0-10 \int\limits_{1}^{e}(\ln x)^{9} d x$
$=e-10 I_{9}+10 I_{9}=e$
Was this answer helpful?
0
0
Top Questions on integral
- The value of the definite integral \( \int_0^{\pi} \sin^2 x \, dx \) is:
- Evaluate the integral: \[ \int \frac{\sin(2x)}{\sin(x)} \, dx \]
- Find the value of the integral: \[ \int_0^\pi \sin^2(x) \, dx. \]
- Evaluate the integral: $$ \int_0^{\pi/4} \frac{\ln(1 + \tan x)}{\cos x \sin x} \, dx $$
- Evaluate the integral \( \int_0^1 \frac{\ln(1 + x)}{1 + x^2} \, dx \)
View More Questions
Questions Asked in BITSAT exam
- A particle moves along the x-axis under a force $ F(x) = 6x^2 $ N. The work done by this force in moving the particle from $ x = 1 \, \text{m} $ to $ x = 2 \, \text{m} $ is:
- BITSAT - 2025
- work, energy and power
- If $ f(x) = \sin^{-1}(2x\sqrt{1 - x^2}) $, then $ f'(x) $ is:
- BITSAT - 2025
- Inverse Trigonometric Functions
- Find the angle between the vectors \( \mathbf{a} = (2, 3, 1) \) and \( \mathbf{b} = (1, -1, 4) \).
- BITSAT - 2025
- Vector Algebra
- Two identical bodies of mass 1 kg each are moving towards each other with velocities of 5 m/s and 3 m/s, respectively. They collide elastically. What will be the velocity of the body initially moving at 5 m/s after the collision?
- BITSAT - 2025
- Mechanics
- In the reaction \( \mathrm{N_2(g)} + 3\mathrm{H_2(g)} \leftrightarrow 2\mathrm{NH_3(g)} \), if the equilibrium constant \( K_c = 4 \times 10^{-3} \) at a certain temperature, which of the following is true about the reaction at equilibrium?
- BITSAT - 2025
- Equilibrium
View More Questions