To determine

To evaluate:

The given definite integral ∫12xx-1 dx

Answer

1615

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 g’(x)dx =du

ii) Indefinite integral

∫xn dx=xn+1n+1+C   (n≠-1)

iii)

∫ab[fx+gx] dx=∫abfxdx+∫abgxdx

2) Given:

The given integral is

∫12xx-1 dx.

3) Calculations:

Here, use the substitution method.

Substitute x-1=u, then   x=u+1.

Differentiating with respect to x

dx=du

The limits change and the new limits of integration are calculated by substituting

for x=1, u=1-1=0 & for x=2, u=2-1 =1

Therefore, the given integral becomes

∫12xx-1 dx=∫01(u+1)u du

=∫01u+1u1/2 du

=∫01u3/2+u1/2 du

=∫01u3/2 du+∫01u1/2 du

Integrating

=u32+132+101+u12+112+101

Simplifying this, we get

=u5/25/201+u3/23/201

Substituting the limits.

=25152-052+23132-032

Solving this, we get

=25(1-0)+23(1-0)

=25+23

=1615

Conclusion:

Therefore,

∫12xx-1 dx=1615