Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
NDA 2015
Exam
NDA
Year
2015
Subject
Mathematics
Chapter
Indefinite Integration

NDA 2015 · Mathematics · Indefinite Integration

∫ dx 1+ ^ -x is equal to where c is the constant of integration

∫dx1+\mathrme−x\int \frac{\mathrm{dx}}{1+\mathrme^{-\mathrm{x}}} is equal to where cc is the constant of integration

Options

  1. A1+\mathrmex+c1+\mathrme^{\mathrm{x}}+\mathrm{c}
  2. B\ln (1+\mathrme−x)+c\left(1+\mathrme^{-\mathrm{x}}\right)+\mathrm{c}
  3. C\ln (1+\mathrmex)+c\left(1+\mathrme^{\mathrm{x}}\right)+\mathrm{c}
  4. D2 \ln (1+\mathrme−x)+c\left(1+\mathrme^{-\mathrm{x}}\right)+\mathrm{c}

Correct answer

C. \ln (1+\mathrmex)+c\left(1+\mathrme^{\mathrm{x}}\right)+\mathrm{c}

Step-by-step solution

∫dx1+e−x⇒∫exex+1dxLet ex+1=texdx=dt=∫dtt⇒log⁡t+c⇒log⁡(ex+1)+c\begin{aligned} & \int \frac{\mathrm{dx}}{1+\mathrm{e}^{-\mathrm{x}}} \\ & \Rightarrow \int \frac{\mathrm{e}^{\mathrm{x}}}{\mathrm{e}^{\mathrm{x}}+1} \mathrm{dx} \\ & \text{Let } \mathrm{e}^{\mathrm{x}}+1=\mathrm{t} \\ & \mathrm{e}^{\mathrm{x}} \mathrm{dx}=\mathrm{dt} \\ & =\int \frac{\mathrm{dt}}{\mathrm{t}} \\ & \Rightarrow \log \mathrm{t}+\mathrm{c} \\ & \Rightarrow \log \left(\mathrm{e}^{\mathrm{x}}+1\right)+\mathrm{c}\end{aligned}
Practice Indefinite Integration on Quantrex Academy →

More from Indefinite Integration

Passage: Let ∫( θ\,dθ)/((2+ θ)(3+4 θ))=A |2+ θ|+B |3+4 θ| . Question: What is the value… NDA 2026Passage: Let ∫( θ\,dθ)/((2+ θ)(3+4 θ))=A |2+ θ|+B |3+4 θ| . Question: What is the value… NDA 2026What is ∫ dx 10^x + 10^ -x equal to? NDA 2025Consider the following for the two (02) items that follow: Let the function y = (1 - x)^… NDA 2025For the following two (02) items: Let u = ∫ e^x x\, dx and v = ∫ e^x x\, dx . Consider… NDA 2025For the following two (02) items: Let u = ∫ e^x x\, dx and v = ∫ e^x x\, dx . What is u +… NDA 2025Let ∫ d x x+1 - x-1 =α(x+1)^ (3)/(2) +β(x-1)^ (3)/(2) +c What is the value of β ? NDA 2024Paragraph: Let 2 ∫ x^2-1 x^2+1 d x=U(x) V(x)-3 \ U(x)+V(x)\ +c Question: What is U(x)… NDA 2024Paragraph: Let 2 ∫ x^2-1 x^2+1 d x=U(x) V(x)-3 \ U(x)+V(x)\ +c Question: What is… NDA 2024Given that ∫ (3 x+4 x)/(2 x+5 x) d x=(α x)/(29)+(β)/(29) |2 x+5 x|+c What is the value of… NDA 2024Given that ∫ (3 x+4 x)/(2 x+5 x) d x=(α x)/(29)+(β)/(29) |2 x+5 x|+c What is the value of… NDA 2024Let ∫ d x x+1 - x-1 =α(x+1)^ (3)/(2) +β(x-1)^ (3)/(2) +c What is the value of α ? NDA 2024Paragraph: Let f(x)=P e^x+Q e^ 2 x +R e^ 3 x , where P, Q, R are real numbers. Further… NDA 2023Paragraph: Let f(x)=P e^x+Q e^ 2 x +R e^ 3 x , where P, Q, R are real numbers. Further… NDA 2023What is ∫(x^x)^2(1+ x) d x equal to? NDA 2022What is ∫ e^ x (1+ x+x x) d x equal to? NDA 2022 Full Indefinite Integration list Indefinite Integration JEE Maths PYQsJEE Mathematics PYQs All NDA PYQs