int {frac{{a{x^3} + b{x^2} + c}}{{{x^4}}},dx} | vedclass

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(C) Given integral: $I = \int {\frac{{a{x^3} + b{x^2} + c}}{{{x^4}}}dx}$ Divide each term in the numerator by the denominator: $I = \int {\left( {\fra

Vedclass · Accepted Answer

$a\log |x| - \frac{b}{x} - \frac{c}{3{x^3}} + C$

Vedclass · Answer

(C) Given integral: $I = \int {\frac{{a{x^3} + b{x^2} + c}}{{{x^4}}}dx}$ Divide each term in the numerator by the denominator: $I = \int {\left( {\frac{{a{x^3}}}{{{x^4}}} + \frac{{b{x^2}}}{{{x^4}}} + \frac{c}{{{x^4}}}} \right)dx}$ $I = \int {\left( {\frac{a}{x} + \frac{b}{{{x^2}}} + \frac{c}{{{x^4}}}} \right)dx}$ $I = \int {\frac{a}{x}dx + \int {b{x^{ - 2}}dx + \int {c{x^{ - 4}}dx} } }$ Using the integration formulas $\int {\frac{1}{x}dx = \log |x|}$ and $\int {{x^n}dx = \frac{{{x^{n + 1}}}}{{n + 1}}}$: $I = a\log |x| + b\left( {\frac{{{x^{ - 1}}}}{{ - 1}}} \right) + c\left( {\frac{{{x^{ - 3}}}}{{ - 3}}} \right) + C$ $I = a\log |x| - \frac{b}{x} - \frac{c}{{3{x^3}}} + C$ Thus,the correct option is $C$.

$\int {\frac{{a{x^3} + b{x^2} + c}}{{{x^4}}}\,dx} $ equals to

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