\(\int_{0}^{1}\displaystyle x^2e^{x^3}\,dx\) is equal to:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle\frac{1}{3}(e-1)\)
The factor \(x^2\) is, up to the constant 3, the derivative of the power \(x^3\), so substitute \(t = x^3\):
\(dt = 3x^2\,dx\)
\(x = 0 \Rightarrow t = 0, \quad x = 1 \Rightarrow t = 1\)
Then
\( \displaystyle \int_{0}^{1}x^2e^{x^3}\,dx = \frac{1}{3}\int_{0}^{1}e^{t}\,dt \)
\( \displaystyle = \frac{1}{3}\Big[e^{t}\Big]_{0}^{1} \)
\( \displaystyle = \frac{1}{3}(e-1) \)