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1

Solved example of Improper integrals

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

Solve the integral $\int\frac{\sqrt{y^{7}}-\sqrt[3]{y^{5}}-\sqrt[4]{y}}{y^2}dy$ applying u-substitution. Let $u$ and $du$ be

$\begin{matrix}u=\sqrt[4]{y} \\ du=\frac{1}{4}y^{-\frac{3}{4}}dy\end{matrix}$
3

Isolate $dy$ in the previous equation

$\frac{du}{\frac{1}{4}y^{-\frac{3}{4}}}=dy$
4

Rewriting $y$ in terms of $u$

$y=u^{4}$
5

Substituting $u$, $dy$ and $y$ in the integral and simplify

$\int\frac{u^{14}-\sqrt[3]{u^{20}}-u}{\frac{1}{4}u^{5}}du$
6

Take the constant out of the integral

$4\int\frac{u^{14}-\sqrt[3]{u^{20}}-u}{u^{5}}du$
7

Split the fraction $\frac{u^{14}+-\sqrt[3]{u^{20}}-u}{u^{5}}$ in two terms with common denominator ($u^{5}$)

$4\int\left(\frac{u^{14}}{u^{5}}+\frac{-\sqrt[3]{u^{20}}-u}{u^{5}}\right)du$
8

Split the fraction $\frac{-\sqrt[3]{u^{20}}-u}{u^{5}}$ in two terms with common denominator $u^{5}$

$4\int\left(\frac{u^{14}}{u^{5}}+\frac{-\sqrt[3]{u^{20}}}{u^{5}}+\frac{-u}{u^{5}}\right)du$
9

Simplifying the fraction by $u$

$4\int\left(\frac{u^{14}}{u^{5}}+\frac{-\sqrt[3]{u^{20}}}{u^{5}}+\frac{-1}{u^{4}}\right)du$
10

Simplifying the fraction by $u$

$4\int\left(u^{9}-\sqrt[3]{u^{5}}+\frac{-1}{u^{4}}\right)du$
11

The integral of the sum of two or more functions is equal to the sum of their integrals

$4\int u^{9}du+4\int-\sqrt[3]{u^{5}}du+4\int\frac{-1}{u^{4}}du$
12

The integral of a constant by a function is equal to the constant multiplied by the integral of the function

$4\int u^{9}du-4\int\sqrt[3]{u^{5}}du+4\int\frac{-1}{u^{4}}du$
13

Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a constant function

$4\int u^{9}du-\frac{3}{2}\sqrt[3]{u^{8}}+4\int\frac{-1}{u^{4}}du$
14

Substitute $u$ back for it's value, $\sqrt[4]{y}$

$4\int u^{9}du-\frac{3}{2}\sqrt[3]{\left(\sqrt[4]{y}\right)^{8}}+4\int\frac{-1}{u^{4}}du$
15

Applying the power of a power property

$4\int u^{9}du-\frac{3}{2}\sqrt[3]{y^{2}}+4\int\frac{-1}{u^{4}}du$
16

Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a constant function

$\frac{4u^{10}}{10}-\frac{3}{2}\sqrt[3]{y^{2}}+4\int\frac{-1}{u^{4}}du$
17

Substitute $u$ back for it's value, $\sqrt[4]{y}$

$\frac{4\left(\sqrt[4]{y}\right)^{10}}{10}-\frac{3}{2}\sqrt[3]{y^{2}}+4\int\frac{-1}{u^{4}}du$
18

Applying the power of a power property

$\frac{4\sqrt{y^{5}}}{10}-\frac{3}{2}\sqrt[3]{y^{2}}+4\int\frac{-1}{u^{4}}du$
19

Take $\frac{4}{10}$ out of the fraction

$\frac{2}{5}\sqrt{y^{5}}-\frac{3}{2}\sqrt[3]{y^{2}}+4\int\frac{-1}{u^{4}}du$
20

Rewrite the exponent using the power rule $\frac{a^m}{a^n}=a^{m-n}$, where in this case $m=0$

$\frac{2}{5}\sqrt{y^{5}}-\frac{3}{2}\sqrt[3]{y^{2}}+4\int-u^{-4}du$
21

The integral of a constant by a function is equal to the constant multiplied by the integral of the function

$\frac{2}{5}\sqrt{y^{5}}-\frac{3}{2}\sqrt[3]{y^{2}}-4\int u^{-4}du$
22

Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a constant function

$\frac{2}{5}\sqrt{y^{5}}-\frac{3}{2}\sqrt[3]{y^{2}}+\frac{-4u^{-3}}{-3}$
23

Substitute $u$ back for it's value, $\sqrt[4]{y}$

$\frac{2}{5}\sqrt{y^{5}}-\frac{3}{2}\sqrt[3]{y^{2}}+\frac{-4\left(\sqrt[4]{y}\right)^{-3}}{-3}$
24

Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number

$\frac{2}{5}\sqrt{y^{5}}-\frac{3}{2}\sqrt[3]{y^{2}}+\frac{\frac{-4}{\left(\sqrt[4]{y}\right)^{3}}}{-3}$
25

Applying the power of a power property

$\frac{2}{5}\sqrt{y^{5}}-\frac{3}{2}\sqrt[3]{y^{2}}+\frac{\frac{-4}{\sqrt[4]{y^{3}}}}{-3}$
26

Simplifying the fraction

$\frac{2}{5}\sqrt{y^{5}}-\frac{3}{2}\sqrt[3]{y^{2}}+\frac{-4}{-3\sqrt[4]{y^{3}}}$
27

Take out the constant $-3$ from the fraction's denominator

$\frac{2}{5}\sqrt{y^{5}}-\frac{3}{2}\sqrt[3]{y^{2}}-\frac{1}{3}\left(\frac{-4}{\sqrt[4]{y^{3}}}\right)$
28

Multiplying the fraction and term

$\frac{2}{5}\sqrt{y^{5}}-\frac{3}{2}\sqrt[3]{y^{2}}+\frac{\frac{4}{3}}{\sqrt[4]{y^{3}}}$
29

As the integral that we are solving is an indefinite integral, when we finish we must add the constant of integration

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

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