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Step-by-step Solution

Integral of $\frac{y^{\left(\frac{7}{2}\right)}-y^{\left(\frac{5}{3}\right)}-y^{\left(\frac{1}{4}\right)}}{y^2}$ with respect to x

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Answer

$\frac{2}{5}\sqrt{y^{5}}-\frac{3}{2}\sqrt[3]{y^{2}}+\frac{4}{3}y^{-\frac{3}{4}}+C_0$

Step-by-step explanation

Problem to solve:

$\int\frac{y^{\frac{7}{2}}-y^{\frac{5}{3}}-y^{\frac{1}{4}}}{y^2}dy$
1

Split the fraction $\frac{\sqrt{y^{7}}+-\sqrt[3]{y^{5}}-\sqrt[4]{y}}{y^2}$ in two terms with same denominator ($y^2$)

$\int\left(\frac{\sqrt{y^{7}}}{y^2}+\frac{-\sqrt[3]{y^{5}}-\sqrt[4]{y}}{y^2}\right)dy$
2

Split the fraction $\frac{-\sqrt[3]{y^{5}}+-\sqrt[4]{y}}{y^2}$ in two terms with same denominator ($y^2$)

$\int\left(\frac{\sqrt{y^{7}}}{y^2}+\frac{-\sqrt[3]{y^{5}}}{y^2}+\frac{-\sqrt[4]{y}}{y^2}\right)dy$

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Answer

$\frac{2}{5}\sqrt{y^{5}}-\frac{3}{2}\sqrt[3]{y^{2}}+\frac{4}{3}y^{-\frac{3}{4}}+C_0$
$\int\frac{y^{\frac{7}{2}}-y^{\frac{5}{3}}-y^{\frac{1}{4}}}{y^2}dy$

Main topic:

Integrals of Rational Functions

Time to solve it:

~ 1.12 seconds