Question

Find the integration of \(\frac{{{{\rm{x}}^4} + 3{{\rm{x}}^2} - {\rm{x}} + 5}}{{\rm{x}}}\)
1. \(\frac{{{x^4}}}{4} + \frac{{3{x^2}}}{2} + x + \;log{x^5} + c\)
2. \(\frac{{{x^4}}}{4} + \frac{{3{x^2}}}{2} - x + \;log{x^5} + c\)
3. \(\frac{{{x^4}}}{4} + \frac{{5{x^2}}}{2} + x + \;log{x^5} + c\)
4. \(\frac{{{x^5}}}{5} + \frac{{3{x^2}}}{2} - x + \;log{x^5} + c\)

Answer 1

Correct Answer - Option 2 : \(\frac{{{x^4}}}{4} + \frac{{3{x^2}}}{2} - x + \;log{x^5} + c\)

Concept:

\(\smallint \frac{1}{x}\;dx = \log x\)

\(\smallint {x^n}\;dx = \;\frac{{{x^{n\; + \;1}}}}{{n + 1}} + c\)

n log m = log mⁿ

Calculation:

\(\frac{{{{\rm{x}}^4} + 3{{\rm{x}}^2} - {\rm{x}} + 5}}{{\rm{x}}}\)

= \(\frac{{{x^4}}}{x} + \;\frac{{3{x^2}}}{x} - \;\frac{x}{x} + \;\frac{5}{x}\)

= x³ + 3x – 1 + \(\frac{5}{x}\)

Now, I = ∫ \(\frac{{{{\rm{x}}^4} + 3{{\rm{x}}^2} - {\rm{x}} + 5}}{{\rm{x}}}\)dx

= \(\smallint ({x^3} + \;3x - 1 + \;\frac{5}{x})\;dx\)

= \(\smallint {x^3}\;dx + 3\smallint xdx - \;\smallint 1\;dx + \;\smallint \frac{5}{x}\;dx\)

= \(\frac{{{x^4}}}{4}\; + \;3\frac{{{x^2}}}{2}\; - \;x\; + \;5\;logx\) + c

= \(\frac{{{x^4}}}{4} + \frac{{3{x^2}}}{2} - x + \;log{x^5} + c\)