What is the integral of (1 + e^(-x))^2 dx?

Prepare for the JEE Main Integration Test with interactive quizzes and detailed explanations. Boost your integration skills, understand complex problems, and ace your exam. Master the dynamics of integration and put your knowledge to the test!

To find the integral of ( (1 + e^{-x})^2 , dx ), we can start by expanding the integrand:

[

(1 + e^{-x})^2 = 1 + 2e^{-x} + e^{-2x}

]

Now we can integrate each term separately:

The integral of ( 1 ) with respect to ( x ) is ( x ).
The integral of ( 2e^{-x} ) can be found using the formula for the integral of the exponential function. The integral is:

[

\int 2e^{-x} , dx = -2e^{-x}

]

[

\int e^{-2x} , dx = -\frac{1}{2} e^{-2x}

]

Putting all of this together, we have:

[

\int (1 + e^{-x})^2 , dx = \int (1 + 2e^{-x} + e^{-2x}) , dx = x - 2e^{-x} - \frac{1}{