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akutagawatj

Answered question

2021-12-07

Find indefinite integral.

\int (x^{\frac{1}{2}} + 3 x^{- \frac{2}{3}}) d x

Charles Clute

Beginner2021-12-08Added 17 answers

Step 1
Let the given integral be y −

y = \int (x^{\frac{1}{2}} + 3 x^{- \frac{2}{3}}) d x

Identity to be used −

\int x^{n} d x = \frac{x^{(n + 1)}}{n + 1} + C

Step 2
Using identity in the given function −

y = [\frac{x^{(\frac{1}{2} + 1)}}{(\frac{1}{2} + 1)}] + 3 [\frac{x^{(- \frac{2}{3} + 1)}}{(- \frac{2}{3} + 1)}] + C

y = [\frac{x^{\frac{3}{2}}}{\frac{3}{2}}] + 3 [\frac{x^{\frac{1}{3}}}{\frac{1}{3}}] + C

y = \frac{2}{3} x^{\frac{3}{2}} + 3 (3) x^{\frac{1}{3}} + C

y = \frac{2}{3} x^{\frac{3}{2}} + 9 x^{\frac{1}{3}} + C

Hence, the integration of given integral is −

\int (x^{\frac{1}{2}} + 3 x^{- \frac{2}{3}}) d x = \frac{2}{3} x^{\frac{3}{2}} + 9 x^{\frac{1}{3}} + C

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