#### To determine

**To evaluate:**

The integral ∫x2x3+1 dx by making the given substitution u=x3+1.

#### Answer

29x3+132+C

#### Explanation

**1) Concept:**

i) The substitution rule

If u=g(x) is a differentiable function whose range is an interval I and f is continuous on I, then ∫f(gx)g'xdx=∫f(u)du

ii) Indefinite integral

∫xn dx=xn+1n+1+C (n≠-1)

**2) Given:**

∫x2x3+1 dx u=x3+1

**3) Calculation:**

Use thesubstitution u=x3+1.

Differentiate u=x3+1 with respect to x.

du=3x2dx

As x2 dx is a part of the integration, solving for x2 dx by dividing both side by 3.

du3=x2dx

By using concept i)

Substitute u=x3+1, du3=x2dx . To write ∫x2x3+1 dx=∫u12du3

By using concept ii)

=13u12+112+1+C

Simplify.

=13u3232+C

Rearranging:

=29u32+C

Resubstitute u=x3+1.

=29x3+132+C

Rearranging;

=29x3+132+C

**Conclusion:**

Therefore,

∫x2x3+1 dx=29x3+132+C