Integrate without substitution

Question

I'm wondering how I could integrate the following without substitution:$$\int \frac{4}{1 + e^{-x}}dx$$

I know we can factor out the constant so that $4 * \int \frac{1}{1 + e^{-x}}$ but I'm stumped as of what to do next. Could anyone help me out?

score 3 · Answer 1 · answered Oct 26 '14 at 20:39

3

HINT: Multiply the fraction by $\dfrac{e^x}{e^x}$.

answered Oct 26 '14 at 20:39

Brian M. Scott

Hi @Brian, I gather that this would give me $4 * \int \frac{e^x}{e^x + 1}$ but I'm not sure how that helps exactly. Could you by any chance explain further? Thanks for taking the time to reply. – user2451412 Oct 27 '14 at 16:23
@user2451412: The numerator of the fraction $\frac{e^x}{e^x+1}$ is the derivative of the denominator, so by inspection it integrates into $\ln(e^x+1)+C$. – Brian M. Scott Oct 27 '14 at 17:20

score 2 · Accepted Answer · answered Oct 26 '14 at 20:43

2

$$\int \frac{1}{1+\frac{1}{e^x}} dx = \int \frac{e^x}{1+ e^x} dx = \ln (1 + e^x) + C $$

answered Oct 26 '14 at 20:43

Aaron Maroja

2 Answers2