Question:
The value of \( \int_{-5}^{-1} \frac{1}{x} \, dx \) is equal to :
The value of \( \int_{-5}^{-1} \frac{1}{x} \, dx \) is equal to :
Show Hint
Always remember the absolute value: \( \int \frac{1}{x} dx = \log|x| \). Without it, you might incorrectly try to evaluate \( \log(-1) \).
Property: \( \log(1/a) = -\log a \).
Property: \( \log(1/a) = -\log a \).
Updated On: Sep 10, 2026
- \( -\log 5 \)
- \( \log 5 \)
- \( \log(-5) \)
- \( 0 \)
Show Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
Concept:
• The integral of \( \frac{1}{x} \) is \( \log |x| \).
• The Fundamental Theorem of Calculus states \( \int_a^b f(x) \, dx = F(b) - F(a) \).
• Always use the absolute value inside the logarithm for real-valued integrals of \( 1/x \).
Step 1: Apply the integration rule for \( 1/x \)
Let the integral be \( I = \int_{-5}^{-1} \frac{1}{x} \, dx \). The antiderivative of \( \frac{1}{x} \) is \( \log |x| \). \[ I = [\log |x|]_{-5}^{-1} \]
Step 2: Substitute the upper and lower limits
\[ I = \log |-1| - \log |-5| \] Since \( |-1| = 1 \) and \( |-5| = 5 \): \[ I = \log 1 - \log 5 \]
Step 3: Simplify the final value
We know that \( \log 1 = 0 \) for any base. \[ I = 0 - \log 5 \] \[ I = -\log 5 \] This matches option (A).
• The integral of \( \frac{1}{x} \) is \( \log |x| \).
• The Fundamental Theorem of Calculus states \( \int_a^b f(x) \, dx = F(b) - F(a) \).
• Always use the absolute value inside the logarithm for real-valued integrals of \( 1/x \).
Step 1: Apply the integration rule for \( 1/x \)
Let the integral be \( I = \int_{-5}^{-1} \frac{1}{x} \, dx \). The antiderivative of \( \frac{1}{x} \) is \( \log |x| \). \[ I = [\log |x|]_{-5}^{-1} \]
Step 2: Substitute the upper and lower limits
\[ I = \log |-1| - \log |-5| \] Since \( |-1| = 1 \) and \( |-5| = 5 \): \[ I = \log 1 - \log 5 \]
Step 3: Simplify the final value
We know that \( \log 1 = 0 \) for any base. \[ I = 0 - \log 5 \] \[ I = -\log 5 \] This matches option (A).
Was this answer helpful?
0
0
Top CBSE CLASS XII Mathematics Questions
- 2, b, c are in A.P. and the range of determinant $\begin{vmatrix}1&1&1\\ 2&b&c\\ 4&b^{2}&c^{2}\end{vmatrix}$ is [2, 16]. Then find range of c is
- A committee of 11 members is to be formed from 8 males and 5 females. Let m denotes the number of ways of selecting committee having atleast 6 males and n denotes the number of ways of selecting atleast 3 females then
Determine whether each of the following relations are reflexive, symmetric, and transitive.
- Relation R in the set A={1,2,3,...13,14} defined as R={(x,y): 3x-y=0}.
- Relation R in the set of N natural numbers defined as R={(x,y): y=x+5 and x<4}.
- Relation R in the set A={1,2,3,4,5,6} defined as R={(x,y): y is divisible by x}.
- Relation R in the set of Z integers defined as R={(x,y): x-y is an integer}
- Relation R in the set of human beings in a town at a particular time given by
- R={(x,y): x and y work at same place}
- R={(x,y): x and y live in the same locality}
- R={(x,y): x is exactly 7 am taller that y}
- R={(x,y): x is wife of y}
- R={(x,y):x is father of y}
Show that the relation R in the set R of real numbers, defined as
R = {(a, b): a ≤ b2 } is neither reflexive nor symmetric nor transitive.Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as
R = {(a, b): b = a + 1} is reflexive, symmetric or transitive.
View More Questions
Top CBSE CLASS XII Methods of Solving First Order, First Degree Differential Equations Questions
The corner points of the feasible region determined by the system of linear constraints are as shown in the following figure:
(i) If \( Z = 3x - 4y \) be the objective function, then find the maximum value of \( Z \).
(ii) If \( Z = px + qy \) where \( p, q>0 \) be the objective function, find the condition on \( p \) and \( q \) so that maximum value of \( Z \) occurs at \( B(4, 10) \) and \( C(6, 8) \).
- The general solution of the differential equation $x \, dy - y \, dx = 0$ is:
- The integrating factor of the differential equation $2x \frac{dy}{dx} - y = 3$ is:
- Solve the following homogeneous differential equation: \( x \frac{dy}{dx} = y - x \sin^2\left(\frac{y}{x}\right) \), given the initial value condition that \( y = \frac{\pi}{6} \) when \( x = 1 \).
- \( \frac{dy}{dx} = F(x, y) \) will be a homogeneous differential equation for which of the following functions ?
(i) \( F(x, y) = 3x + 2y \)
(ii) \( F(x, y) = \sin \frac{y}{x} + \log y - \log x \)
(iii) \( F(x, y) = e^{y/x} + 1 \)
(iv) \( F(x, y) = \sqrt{x^2 + y^2} - y \)
View More Questions
Top CBSE CLASS XII Questions
- 2, b, c are in A.P. and the range of determinant $\begin{vmatrix}1&1&1\\ 2&b&c\\ 4&b^{2}&c^{2}\end{vmatrix}$ is [2, 16]. Then find range of c is
- A boy of mass 50 kg is standing at one end of a, boat of length 9 m and mass 400 kg. He runs to the other, end. The distance through which the centre of mass of the boat boy system moves is
- A capillary tube of radius r is dipped inside a large vessel of water. The mass of water raised above water level is M. If the radius of capillary is doubled, the mass of water inside capillary will be
- A circular disc is rotating about its own axis. An external opposing torque 0.02 Nm is applied on the disc by which it comes rest in 5 seconds. The initial angular momentum of disc is
- A circular disc is rotating about its own axis at uniform angular velocity \(\omega.\) The disc is subjected to uniform angular retardation by which its angular velocity is decreased to \(\frac {\omega}{2}\) during 120 rotations. The number of rotations further made by it before coming to rest is
View More Questions