+56 Some people really try to learn something in here. Please don't troll so often, so we can help people to solve their questions...
My question is this;
\(\int((x^4+1)/(1+x))dx\)
if you find the correct answer, i will send loves xD
Take the integral:
integral (x^4+1)/(x+1) dx
For the integrand (x^4+1)/(x+1), do long division:
= integral (x^3-x^2+x+2/(x+1)-1) dx
Integrate the sum term by term and factor out constants:
= 2 integral 1/(x+1) dx+ integral x^3 dx- integral x^2 dx+ integral x dx- integral 1 dx
For the integrand 1/(x+1), substitute u = x+1 and du = dx:
= 2 integral 1/u du+ integral x^3 dx- integral x^2 dx+ integral x dx- integral 1 dx
The integral of 1/u is log(u):
= 2 log(u)+ integral x^3 dx- integral x^2 dx+ integral x dx- integral 1 dx
The integral of x^3 is x^4/4:
= x^4/4+2 log(u)- integral x^2 dx+ integral x dx- integral 1 dx
The integral of x^2 is x^3/3:
= -x^3/3+x^4/4+2 log(u)+ integral x dx- integral 1 dx
The integral of x is x^2/2:
= x^2/2-x^3/3+x^4/4+2 log(u)- integral 1 dx
The integral of 1 is x:
= 2 log(u)+x^4/4-x^3/3+x^2/2-x+constant
Substitute back for u = x+1:
= x^4/4-x^3/3+x^2/2-x+2 log(x+1)+constant
Factor the answer a different way:
= 1/12 x (3 x^3-4 x^2+6 x-12)+2 log(x+1)+constant
Which is equivalent for restricted x values to:
Answer: | = 1/12 (3 x^4-4 x^3+6 x^2-12 x+24 log(x+1)-25)+constant
