Question:
\( \displaystyle\int \left[ \dfrac{1}{\log x} - \dfrac{1}{(\log x)^{2}} \right] dx \) is equal to :
\( \displaystyle\int \left[ \dfrac{1}{\log x} - \dfrac{1}{(\log x)^{2}} \right] dx \) is equal to :
Show Hint
Differentiate x/log x with the quotient rule and compare with the integrand.
Updated On: Oct 22, 2026
- \( \dfrac{x}{\log x} + C \)
- \( x^{2}\log x + C \)
- \( \dfrac{1}{\log x} + C \)
- \( \dfrac{-1}{(\log x)^{2}} + C \)
Show Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
Step 1: Choose a method:
The integrand does not fit a simple power rule. A good guess for the antiderivative is \( \dfrac{x}{\log x} \), and we prove it by differentiating.
Step 2: Differentiate the guess:
Let \( g(x) = \dfrac{x}{\log x} \). By the quotient rule, \( g^{\prime}(x) = \dfrac{\log x \cdot 1 - x \cdot \dfrac1x}{(\log x)^{2}} \).
Step 3: Simplify:
\[ g^{\prime}(x) = \dfrac{\log x - 1}{(\log x)^{2}} = \dfrac{1}{\log x} - \dfrac{1}{(\log x)^{2}} \]
Step 4: Conclude:
This is exactly the integrand. So the integral is \( \dfrac{x}{\log x} + C \).
Step 5: Check the other options:
Differentiating \( \dfrac{1}{\log x} \) gives \( -\dfrac{1}{x(\log x)^{2}} \), not the integrand. Differentiating \( x^{2}\log x \) gives \( 2x\log x + x \). Differentiating option (D) gives \( \dfrac{2}{x(\log x)^{3}} \). None matches.
Final Answer:
The integral is \( \dfrac{x}{\log x} + C \), option (A). \[ \boxed{\dfrac{x}{\log x} + C} \]
The integrand does not fit a simple power rule. A good guess for the antiderivative is \( \dfrac{x}{\log x} \), and we prove it by differentiating.
Step 2: Differentiate the guess:
Let \( g(x) = \dfrac{x}{\log x} \). By the quotient rule, \( g^{\prime}(x) = \dfrac{\log x \cdot 1 - x \cdot \dfrac1x}{(\log x)^{2}} \).
Step 3: Simplify:
\[ g^{\prime}(x) = \dfrac{\log x - 1}{(\log x)^{2}} = \dfrac{1}{\log x} - \dfrac{1}{(\log x)^{2}} \]
Step 4: Conclude:
This is exactly the integrand. So the integral is \( \dfrac{x}{\log x} + C \).
Step 5: Check the other options:
Differentiating \( \dfrac{1}{\log x} \) gives \( -\dfrac{1}{x(\log x)^{2}} \), not the integrand. Differentiating \( x^{2}\log x \) gives \( 2x\log x + x \). Differentiating option (D) gives \( \dfrac{2}{x(\log x)^{3}} \). None matches.
Final Answer:
The integral is \( \dfrac{x}{\log x} + C \), option (A). \[ \boxed{\dfrac{x}{\log x} + C} \]
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 Indefinite Integrals Questions
- The value of $\displaystyle \int_{0}^{\pi/6} \sin 3x \, dx$ is:
- Evaluate: \[ I = \int_{0}^{\pi/2} \frac{ x \sin x \cos x }{ \sin^4 x + \cos^4 x } \, dx. \]
- Evaluate: \[ \int_{1}^{3} (|x - 1| + |x - 2|) dx. \]
- Find the integral:
\[ I = \int \frac{\cos \theta \, d\theta}{\sqrt{3 - 3\sin \theta - \cos^2 \theta}} \] - Find: \[ I = \int \frac{2x^2 + 1}{x^2(x^2 + 4)} \, dx \]
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