Lets Integrate It #3 !

Calculus Level 4

$\large \int_{0}^{1}\frac{x^{10}-1}{\ln(x)} \, dx=\ ?$ What is the value of given integral upto 3 decimal places ?

1 solution

Tanishq Varshney
Sep 8, 2015

$\large{I(a)=\displaystyle \int^{1}_{0} \frac{x^a-1}{\ln x}dx}$

Differentiating wrt $a$

$\large{I^{\prime} (a)=\displaystyle \int^{1}_{0} \frac{x^{a} \ln x}{\ln x} dx}$

$\large{I^{\prime} (a)=\frac{1}{a+1}}$

$\large{I(a)=\ln (a+1)+c}$

now $I(0)=0$ so $c=0$

$\large{\boxed{I(a)=\ln (a+1)}}$

put $a=10$

$\large{\ln (11)=\boxed{2.397}}$

Why did u differentiate the inside portion, when limits are already there?

Sachin Arora - 5 years, 9 months ago

Instead of explaining my point , i would rather insist u to read this

Tanishq Varshney - 5 years, 9 months ago

Tanishq, would you like to host a wiki collaboration on Differentiate through the Integral ?

Calvin Lin Staff - 5 years, 8 months ago

Thanks bro

Sachin Arora - 5 years, 9 months ago