Key to "Find the most general antiderivative or indefinite integral...."

Armorikam

Answered question

2021-10-15

Find the most general antiderivative or indefinite integral.
$\int (\sqrt[13]{x} + \sqrt[14]{x}) d x$ ?

Answer & Explanation

Nathanael Webber

Skilled2021-10-16Added 117 answers

Step 1
Make two separate integrals
$\int (\sqrt[13]{x} + \sqrt[14]{x}) d x$
$= \int (x^{\frac{1}{13}} + x^{\frac{1}{14}}) d x$
$= \int (x^{\frac{1}{13}}) d x + \int (x^{\frac{1}{14}}) d x$
Step 2
Integrate each integral. Used integration of $x^{n} = x^{\frac{n + 1}{n + 1}}$
C=constant
Answer: $\frac{13}{14} x^{\frac{14}{13}} + \frac{14}{15} x^{\frac{15}{14}} + C$
$\int (x^{\frac{1}{13}}) d x + \int (x^{\frac{1}{14}}) d x$
$= \frac{x^{\frac{1}{13} + 1}}{\frac{1}{13} + 1} + \frac{x^{\frac{1}{14} + 1}}{\frac{1}{14} + 1} + C$
$= \frac{x^{\frac{14}{13}}}{\frac{14}{13}} + \frac{x^{\frac{15}{14}}}{\frac{15}{14}} + C$
$= \frac{13 x^{\frac{14}{13}}}{14} + \frac{14 x^{\frac{15}{14}}}{15} + C$

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