Question

$\displaystyle \sf\int_{0}^{ \infty } \frac{x^{2020} }{1 + x + {x}^{2} + \cdot\cdot\cdot \cdot\cdot\cdot + {x}^{2020} } \: dx$​​

Answer 1

Answer:

At a first glance, I notice that 11+x2 is the derivative of arctan(x). Meanwhile, partial fraction will not work in this case (at least from my perspective) because the power of the x in 11+x2020 is too large. Then I tried to approach this problem by integration by part. Then I got the following result:

∫∞01(1+x2)(1+x2020)dx=tan−1(x)1+x2020|∞0−∫∞0−2020tan−1(x)x2019(1+x2020)2dx

Since limx→∞tan−1(x)=1 and limx→∞1+x2020=∞

Then

tan−1(x)1+x2020|∞0=0

However, how should I proceed from

∫∞0−2020tan−1(x)x2019(1+x2020)2dx

Answer 2

Answer:

hey sis do you remember me

Step-by-step explanation:

I am shivangi