Question:
If
\[
\frac{d}{dx}(F(x))=\frac{1}{e^x+1},
\]
then find \(F(x)\), given that
\[
F(0)=\log\left(\frac{1}{2}\right).
\]
If
\[
\frac{d}{dx}(F(x))=\frac{1}{e^x+1},
\]
then find \(F(x)\), given that
\[
F(0)=\log\left(\frac{1}{2}\right).
\]
Show Hint
The trick of multiplying by \( e^{-x} \) turns the denominator into a sum whose derivative is in the numerator.
Alternatively, solve using \( 1 = (e^x + 1) - e^x \) in the numerator.
Alternatively, solve using \( 1 = (e^x + 1) - e^x \) in the numerator.
Updated On: Sep 11, 2026
Show Solution
Verified By Collegedunia
Solution and Explanation
Concept:
• Finding a function from its derivative involves integration.
• \( F(x) = \int F'(x) dx + C \).
• Use substitution for logarithmic integrals.
Step 1: Set up the integral for F(x)
\[ F(x) = \int \frac{1}{e^x + 1} dx \]
Multiply numerator and denominator by \( e^{-x} \):
\[ F(x) = \int \frac{e^{-x}}{1 + e^{-x}} dx \]
Step 2: Perform substitution
Let \( 1 + e^{-x} = u \), then \( -e^{-x} dx = du \implies e^{-x} dx = -du \).
\[ F(x) = \int -\frac{1}{u} du = -\log |u| + C \]
\[ F(x) = -\log(1 + e^{-x}) + C \]
Step 3: Use the initial condition to find C
Given \( F(0) = \log(1/2) \).
\[ -\log(1 + e^0) + C = \log(1/2) \implies -\log 2 + C = \log(1/2) \]
\[ -\log 2 + C = -\log 2 \implies C = 0 \]
Step 4: Write the final function
\[ F(x) = -\log(1 + e^{-x}) = \log\left(\frac{1}{1 + 1/e^x}\right) = \log\left(\frac{e^x}{e^x + 1}\right) \]
• Finding a function from its derivative involves integration.
• \( F(x) = \int F'(x) dx + C \).
• Use substitution for logarithmic integrals.
Step 1: Set up the integral for F(x)
\[ F(x) = \int \frac{1}{e^x + 1} dx \]
Multiply numerator and denominator by \( e^{-x} \):
\[ F(x) = \int \frac{e^{-x}}{1 + e^{-x}} dx \]
Step 2: Perform substitution
Let \( 1 + e^{-x} = u \), then \( -e^{-x} dx = du \implies e^{-x} dx = -du \).
\[ F(x) = \int -\frac{1}{u} du = -\log |u| + C \]
\[ F(x) = -\log(1 + e^{-x}) + C \]
Step 3: Use the initial condition to find C
Given \( F(0) = \log(1/2) \).
\[ -\log(1 + e^0) + C = \log(1/2) \implies -\log 2 + C = \log(1/2) \]
\[ -\log 2 + C = -\log 2 \implies C = 0 \]
Step 4: Write the final function
\[ F(x) = -\log(1 + e^{-x}) = \log\left(\frac{1}{1 + 1/e^x}\right) = \log\left(\frac{e^x}{e^x + 1}\right) \]
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