✓ Board-verified
Mathematics · 2026 · 3 marks
CBSE 2026 · Region 2 · Set 1 · Q28
If d/(d x)( F (x))=1/(e^x+1), then find F (x) given that F (0)= log 1/2.
If $\displaystyle \frac{\mathrm{d}}{\mathrm{d} x}(\mathrm{~F}(x))=\frac{1}{\mathrm{e}^{x}+1}$, then find $\displaystyle \mathrm{F}(x)$ given that $\displaystyle \mathrm{F}(0)=\log \frac{1}{2}$.
Marking-scheme solution
$\displaystyle \dfrac{d}{dx}(F(x))=\dfrac{1}{\mathrm{e}^{x}+1}$
$\displaystyle \Rightarrow F(x)=\int \dfrac{1}{\mathrm{e}^{x}+1}\, dx$
$\displaystyle =\int \dfrac{\mathrm{e}^{-x}}{1+\mathrm{e}^{-x}}\, dx$
$\displaystyle =-\log\left(1+\mathrm{e}^{-x}\right)+C$
Now, $\displaystyle F(0)=-\log 2+C \Rightarrow \log \dfrac{1}{2}+\log 2=C \Rightarrow C=0$
$\displaystyle \therefore F(x)=-\log\left(1+\mathrm{e}^{-x}\right)$
Note: The above expression is equivalent to $\displaystyle F(x)=x-\log\left(\mathrm{e}^{x}+1\right)$More from Integrals
- Evaluate: ∫_0^π x/(1+ sin x) d x2022 · asked 2×
- Evaluate ∫_-1^1|x^4-x| d x. OR Find ∫ (sin^-1 x)/((1-x^2)^3 / 2) d x.2023
- Find: ∫ (x^2)/((x^2+4)(x^2+9)) d x2023
- Evaluate: ∫_0^π (sin^2026 x)/(sin^2026 x+ cos^2026 x) d x2026
- Evaluate: ∫_1^3 (√x)/(√x+√(4-x)) dx2022
- Find: ∫ ((3 cos x-2) sin x)/(5- sin^2 x-4 cos x) d x OR Evaluate: ∫_-2^2 (x^3+|x|+1)/(x^2+4|x|+4) d x2024
- ∫ (x+5)/((x+6)^2) e^x dx is equal to:2025
- Evaluate ∫ 1/((e^x+e^-x)(e^x-e^-x)) d x log √22023
CBSE Class 12 Mathematics past-paper question from the 2026 board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.