find exact value of \int _0^1\:x^2e^{x^3}\left(1+x^3\right)dx

To find the exact value of the integral \(\int_0^1 x^2 e^{x^3} (1 + x^3) \, dx\), we can simplify it through a substitution. 

First, let's rewrite the integral for clarity:

\[
\int_0^1 x^2 e^{x^3} (1 + x^3) \, dx
\]

We'll use the substitution \( u = x^3 \). Then, \( du = 3x^2 \, dx \), or equivalently, \( \frac{1}{3} du = x^2 \, dx \).

Next, we need to adjust the limits of integration. When \( x = 0 \), \( u = 0^3 = 0 \). When \( x = 1 \), \( u = 1^3 = 1 \). So the new limits for \( u \) will be from 0 to 1.

Substituting these into the integral, we get:

\[
\int_0^1 x^2 e^{x^3} (1 + x^3) \, dx = \int_0^1 e^u (1 + u) \cdot \frac{1}{3} \, du
\]

This simplifies to:

\[
\frac{1}{3} \int_0^1 e^u (1 + u) \, du
\]

Now, we can split this into two integrals:

\[
\frac{1}{3} \left( \int_0^1 e^u \, du + \int_0^1 u e^u \, du \right)
\]

We solve these integrals separately. 

The first integral:

\[
\int_0^1 e^u \, du = e^u \bigg|_0^1 = e - 1
\]

For the second integral, we use integration by parts where \( v = u \) and \( dw = e^u \, du \). Then, \( dv = du \) and \( w = e^u \).

Using integration by parts:

\[
\int u e^u \, du = u e^u \bigg|_0^1 - \int e^u \, du
\]

Evaluating at the bounds and simplifying:

\[
\left. u e^u \right|_0^1 - \int_0^1 e^u \, du = (1 \cdot e^1 - 0 \cdot e^0) - (e^u \bigg|_0^1) = e - (e - 1) = 1
\]

Putting it all together:

\[
\frac{1}{3} \left( (e - 1) + 1 \right) = \frac{1}{3} e
\]

Thus, the exact value of the integral is:

\[
\boxed{\frac{1}{3} e}
\]

Would you like further details or have any questions?

Here are 5 more questions you might consider:
1. How do you perform integration by parts?
2. Can you explain substitution in integrals in more detail?
3. How do you solve definite integrals with variable limits?
4. What are some common techniques for evaluating complex integrals?
5. Can you solve \(\int x \sin(x^2) \, dx\)?

**Tip:** When using substitution in integrals, always remember to change the limits of integration to match your new variable.

Discover the exact value of the integral ∫₀¹ x² e^(x³) (1 + x³) dx using substitution and integration techniques. This problem involves advanced calculus concepts and is suitable for advanced undergraduate and graduate students.

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