# #int_0^1(x^7-1)/lnxdx# = ?

##### 1 Answer

#### Explanation:

We want to solve

#I=int_0^1(x^7-1)/ln(x)dx#

The indefinite integral involves the exponential- and logarithmic integral, so we will take a different approach

This will be solved by **differentiation under the integral sign**

Introduce a parameter

#I(a)=int_0^1(x^a-1)/ln(x)dx#

Notice, we seek

Differentiate both sides with respect to

#I'(a)=int_0^1d/(da)((x^a-1)/ln(x))dx#

#color(white)(I'(a))=int_0^1x^adx#

#color(white)(I'(a))=[x^(a+1)/(a+1)]_0^1#

#color(white)(I'(a))=1/(a+1)#

Integrate both sides with respect to

#I(a)=int1/(a+1)da=ln(a+1)+C#

But

#0=ln(0+1)+C=>C=0#

Thus

#I(a)=ln(a+1)#

For

#I(7)=ln(7+1)=ln(8)#

Or equivalent

#I=int_0^1(x^7-1)/ln(x)dx=ln(8)#