How do you integrate 1/x+x^3?

Question

Please show the steps so I can better understand the process.

Answer 1

Did you mean integrate 1/(x + x^3) ?

1/(x + x^3)=1/(x(1+x^2))=(A/x)-(Bx+C)/(1+x^2)
(Ax^2+A-Bx^2-Cx)/(x+x^3)=1/(x+x^3)
(A-B)x^2-Cx+A=1
A-B=0 → A=B
C=0
A=1 → B=1

1/(x + x^3)=(1/x)-x/(x^2+1)
integrate dx/(x + x^3)=integrate(dx/x) - integrate(xdx/(x^2+1))
=lnx - 1/2ln(x^2+1)+c

x^2+1=u
2xdx=du
integrate(xdx/(x^2+1))=1/2integrate(2xdx/(x^2+1))
=1/2integrate(du/u)=1/2ln(u)+c

Answer 2

ln x + (x^4)/4 + C

Not many steps to show.
Primitive of 1/x is ln x, standard result. In fact it's often used as the definition of ln x.

Again, standard rule for integrating a power of x, such as x^3, is
add 1 to the exponent and divide by the new exponent.
Thus integral of x^n
= (x^(n+1))/(n+1) provided n is not equal to -1.

Answer 3

It depends on whether the x + x^3 is all part of the denominator. Did you mean integrate 1/(x + x^3) ? If so then this is nowhere near as easy. The substitution x = tanA might help but I haven't time to work all the way through it.

Answer 4

integral(1/x + x^3)
= integral(1/x) + integral(x^3)
= ln|x| + C + x^4/4 + C
= ln|x| + x^4/4 + C

Answer 5

wow maths not my subject