#### To determine

**To evaluate:**

The given definite integral.

#### Answer

* 0*

#### 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)

∫exdx=ex+c

iii)

∫aafxdx=0

**2) Given:**

∫02x-1ex-12dx

**3) Calculation:**

The given integral is

∫02x-1ex-12dx

Here using the substitution method

Substitute x-12=u

Differentiating with respect to x

2x-1dx=du

x-1dx=du2

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

For x=0, u=0-12=1, and for x=2, u=2-12=1

Therefore, the given integral becomes

∫02x-1ex-12dx=∫11eu12du

But,∫aafxdx=0

So,

∫11eu12du=0

Therefore,

∫02x-1ex-12dx=0

**Conclusion:**

Therefore,

∫02x-1ex-12dx=0