2021-12-31

Determine the following indefinite integral.
$\int \left(\sqrt[4]{{x}^{3}}+\sqrt{{x}^{5}}\right)dx$

Cheryl King

Step 1
To evaluate the indefinite integral, $\int \left(\sqrt[4]{{x}^{3}}+\sqrt{{x}^{5}}\right)dx$.
Solution:
The given integral is,
$\int \left(\sqrt[4]{{x}^{3}}+\sqrt{{x}^{5}}\right)dx$
Simplifying the given integral gives,
$\int \left(\sqrt[4]{{x}^{3}}+\sqrt{{x}^{5}}\right)dx=\int \left({x}^{3/4}+{x}^{5/2}\right)dx$
$=\int {x}^{\frac{3}{4}}dx+\int {x}^{\frac{5}{2}}dx$
Step 2
Solving the given integral further we get,
$\int {x}^{\frac{3}{4}}dx+\int {x}^{\frac{5}{2}}dx=\frac{{x}^{\frac{7}{4}}}{\frac{7}{4}}+\frac{{x}^{\frac{7}{2}}}{\frac{7}{2}}+c$
$=\frac{4}{7}{x}^{\frac{7}{4}}+\frac{2}{7}{x}^{\frac{7}{2}}+c$
Hence, the value of the integral is $\frac{4}{7}{x}^{74}+27{x}^{72}+c$.

Terry Ray

$\int \left(\sqrt{{x}^{5}}+\sqrt[4]{{x}^{3}}\right)dx$
$\int \left({x}^{\frac{5}{2}}+{x}^{\frac{3}{4}}\right)dx$
$=\int {x}^{\frac{5}{2}}dx+\int {x}^{\frac{3}{4}}dx$
$\int {x}^{\frac{5}{2}}dx$
$=\frac{2{x}^{\frac{7}{2}}}{7}$
$\int {x}^{\frac{3}{4}}dx$
$=\frac{4{x}^{\frac{7}{4}}}{7}$
$\int {x}^{\frac{5}{2}}dx+\int {x}^{\frac{3}{4}}dx$
$=\frac{2{x}^{\frac{7}{2}}}{7}+\frac{4{x}^{\frac{7}{4}}}{7}$
$\int \left({x}^{\frac{5}{2}}+{x}^{\frac{3}{4}}\right)dx$
$=\frac{2{x}^{\frac{7}{2}}}{7}+\frac{4{x}^{\frac{7}{4}}}{7}+C$
Let's rewrite / simplify:
$=\frac{2\left({x}^{\frac{7}{2}}+2{x}^{\frac{7}{4}}\right)}{7}+C$

karton

$\begin{array}{}\int \sqrt[4]{{x}^{3}}+\sqrt{{x}^{5}}dx\\ \int {x}^{\frac{3}{4}}+{x}^{2}\sqrt{x}dx\\ \int {x}^{\frac{3}{4}}+{x}^{2}×{x}^{\frac{1}{2}}dx\\ \int {x}^{\frac{3}{4}}+{x}^{\frac{5}{2}}dx\\ \int {x}^{\frac{3}{4}}dx+\int {x}^{\frac{5}{2}}dx\\ \frac{4x\sqrt[4]{{x}^{3}}}{7}+\frac{2{x}^{3}\sqrt{x}}{7}\\ \frac{4x\sqrt[4]{{x}^{3}}+2{x}^{3}\sqrt{x}}{7}\\ \frac{4x\sqrt[4]{{x}^{3}}+2{x}^{3}\sqrt{x}}{7}+C\end{array}$