Key to "Find the indefinite integral. ∫1x23(1+x13)dx"

ddaeeric

Answered question

2021-11-07

Find the indefinite integral.

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

liingliing8

Skilled2021-11-08Added 95 answers

Step 1
Consider the integral

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

Step 2
Use the substitution in the given integral.

u = x^{\frac{1}{3}} and d u = \frac{1}{3} x^{- \frac{2}{3}} d x

3 d u = \frac{d x}{x^{\frac{2}{3}}}

Step 3
Integrate the given function with respect to x.

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

= 3 \int \frac{1}{(1 + u)} d u

3 \ln (1 + u) + c

= 3 \ln (1 + x^{\frac{1}{3}}) + c

Step 4
Thus, the value of the given integral is

3 \ln (1 + x^{\frac{1}{3}}) + c

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