#### To determine

**To evaluate:**

The given integral.

#### Answer

231+ex32+c

#### Explanation

**1) Concept:**

The substitution rule: If u=g(x) is a differentiable function whose range is I, and f is continuous on I, then ∫f(gx)g'xdx=∫f(u)du. Here g(x) is substituted as u and then g’(x)dx =du

**2) Formula:**

i.

∫cf(x)dx=c∫f(x)dx

ii.

∫xndx=xn+1n+1

**3) Given:**

∫ex1+ex dx

**4) Calculation:**

Consider the given integral ∫ex1+ex dx

Here, we use the substitution method

Substitute 1+ex=u,

Differentiating with respect to x

ex dx=du,

Using this in the given integral, it becomes

∫ex1+exdx= ∫udu

By using rules of exponent it can be written as

= ∫u12du

By using the power rule of indefinite integral

=u3232+c

=2 u323+c

Resubstituting u=1+ex in the above solution

=231+ex32+c

**Conclusion:**

Therefore, ∫ex1+exdx= 231+ex32+c