"Evaluate the integral. ∫2x2x+1dx" solved and explained

Emily-Jane Bray

Answered question

2021-09-16

Evaluate the integral.

\int \frac{2^{x}}{2^{x} + 1} d x

jlo2niT

Skilled2021-09-17Added 96 answers

Step 1
Consider the integrals:

\int \frac{2^{x}}{2^{x} + 1} d x

To solve the given integrals,
Let,

2^{x} + 1 = t

Differentiating with respect to "t"

2^{x} \ln 2 d x + 0 = d t

2^{x} d x = \frac{d t}{\ln 2}

Step 2
Substitute all value in equation given integrals,

\int \frac{2^{x}}{2^{x} + 1} d x = \int \frac{1}{t} \frac{d t}{\ln 2}

= \frac{1}{\ln 2} \int \frac{1}{t} d t

= \frac{1}{\ln 2} \cdot \ln | t | + C

= \frac{1}{\ln 2} \cdot \ln | 2^{x} + 1 | + C

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