#### To determine

**To evaluate:**

The given definite integral.

#### Answer

*ln(e+1)*

#### Explanation

**1) Concept:**

i) The substitution rule: for definite integral: If g'(x) is a continuous function on a,b whose f is continuous on range of u=g(x), then ∫abfgxg'(x)dx=∫g(a)g(b)f(u)du. Here g(x) is substituted as u and then differentiation g’(x)dx =du

ii)

∫1xdx=lnx+c

**2) Given:**

∫01ez+1ez+zdz

**3) Calculation:**

The given integral is

∫01ez+1ez+zdz

Here using the substitution method

Substitute ez+z=u

Differentiating with respect to z

(ez+1) dz=du,

The limits changes; the new limits of integration are calculated by substituting

For z=0, u=e0+0=1, and for z=1, u=e1+1=e+1

Therefore, the given integral becomes

∫01ez+1ez+zdz=∫1e+11udu

Integrating this by using the fundamental theorem of calculus and power rule of integration

*=lnu1e+1*

=ln|e+1|-ln|1|

=lne+1-0

Since always e+1>0

=ln(e+1)

**Conclusion:**

Therefore,

∫01ez+1ez+zdz=ln(e+1)