Answer to Question #131180 in Calculus for fumi

Question

Integral Calculus
1. ∫ x^4-2x^3+3x^2-x+3/x^3-2x^2+3x dx

Answer 1

ANSWER

$\dfrac {1}{5}x^5 - \dfrac {1}{2}x^4 + \dfrac {1}{3}x^3 + x^2 - \dfrac {3}{2x^2} + K$

SOLUTION

$\int (x^4 - 2x^3 + 3x^2 - x + \dfrac {3}{x^3} - 2x^2 + 3x) dx$

Simplifying the expression by collecting like terms:

$= \int [x^4 - 2x^3 + (3x^2 - 2x^2) + (3x - x) + 3x^{-3}]dx$

$= \int (x^4 - 2x^3 +x^2 + 2x + 3x^{-3}) dx$

Solving the integral gives:

$= \dfrac {x^5}{5} - \dfrac {2x^4}{4} + \dfrac {x^3}{3} + \dfrac {2x^2}{2} + \dfrac {3x^{-2}}{-2} + K$

$= \dfrac {1}{5}x^5 - \dfrac {1}{2}x^4 + \dfrac {1}{3}x^3 + x^2 - \dfrac {3}{2x^2} + K$