Question #2ec64

2 Answers
Apr 16, 2017

# x^2/2+x-2/sqrt5ln|(2x+1-sqrt5)/(2x+1+sqrt5)|+C.#

Explanation:

Let, #I=int(x^3+2x^2-3)/(x^2+x-1)dx#

By Long Division, we get,

#(x^3+2x^2-3)/(x^2+x-1)=x+1-2/(x^2+x-1),# so,

#I=int(x+1)dx-2int1/(x^2+x-1)dx#

#=x^2/2+x-2int1/{x^2+x+1/4-5/4)dx#

#=x^2/2+x-2int1/{(x+1/2)^2-(sqrt5/2)^2} dx#

But, we know that, #int1/(x^2-a^2)=1/(2a)ln|(x-a)/(x+a)|+c.,#

#:., I=x^2/2+x-2/(2*sqrt5/2)ln|(x+1/2-sqrt5/2)/(x+1/2+sqrt5/2)|,i.e.,#

#I=x^2/2+x-2/sqrt5ln|(2x+1-sqrt5)/(2x+1+sqrt5)|+C.#

Enjoy Maths.!

Apr 16, 2017

You can factor using the quadratic formula.

Explanation:

You have done this part:

#int(x^3+2x^2-3)/(x^2+x-1)dx= intxdx+intdx-2int1/(x^2+x-1)dx#

Quadratic formula:

#x = (-b+-sqrt(b^2-4(a)(c)))/(2a)#

#x = (-1+-sqrt(1^2-4(1)(-1)))/(2(1))#

#x = (-1+-sqrt(5))/2#

This makes the factors:

#(x+(1+sqrt5)/2)(x+(1-sqrt5)/2)#

Partial Fraction decomposition:

#1/((x+(1+sqrt5)/2)(x+(1-sqrt5)/2)) = A/(x+(1+sqrt5)/2)+ B/(x+(1-sqrt5)/2)#

Multiply by the denominator:

#1 = A(x+(1-sqrt5)/2)+ B(x+(1+sqrt5)/2)#

Eliminate B by letting #x = -(1+sqrt5)/2#

#1 = A(-(1+sqrt5)/2+(1-sqrt5)/2)#

#1 = A(-sqrt5)#

#A = -sqrt5/5#

Eliminate A by letting #x = -(1-sqrt5)/2#

#1 = B(-(1-sqrt5)/2+(1+sqrt5)/2)#

#1 = B(sqrt5)#

#B = sqrt5/5#

The integral becomes:

#int(x^3+2x^2-3)/(x^2+x-1)dx= intxdx+intdx-2sqrt5/5int1/(x+(1-sqrt5)/2)dx + 2sqrt5/5int1/(x+(1+sqrt5)/2)#

The integrals are trivial:

#int(x^3+2x^2-3)/(x^2+x-1)dx= x^2/2+x-2sqrt5/5ln|x+(1-sqrt5)/2| + 2sqrt5/5ln|x+(1+sqrt5)/2| + C#