Explained: "The functions y=ex3 and y=x3ex3 do not have..." - Plainmath

enfurezca3x

Answered question

2021-11-21

The functions

y = e^{x^{3}}

and

y = x^{3} e^{x^{3}}

do not have elementary anti-derivatives, but

y = (1 + 3 x^{3}) e^{x^{3}}

does.
Evaluate

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

.

Answer & Explanation

Otigh1979

Beginner2021-11-22Added 17 answers

Step 1
Given,

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

Step 2
Apply integration by parts,

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

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

= \frac{1 + 3 x^{3}}{3 x^{2}} e^{x^{3}} - 3 \int e^{x^{3}} d x

= \frac{1 + 3 x^{3}}{3 x^{2}} e^{x^{3}} - 3 \cdot \frac{e^{x^{3}}}{3 x^{2}}

= \frac{1 + 3 x^{3}}{3 x^{2}} e^{x^{3}} - \frac{e^{x^{3}}}{x^{2}}

= \frac{1 + 3 x^{3} - 3}{3 x^{2}} e^{x^{3}}

= \frac{3 x^{3} - 2}{3 x^{2}} e^{x^{3}}

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