Using a suitable substitution, [math: ∫02x31+2x2dx] \int_{0}^{2} \frac{x^3}{\sqrt{1+2x^2}} {dx} can be expressed in terms of [math: u] as:

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BlackTom AI · Accepted Answer

The problem asks to express the integral ∫_0^2 x^3 / √(1+2x^2) dx using a suitable substitution, in terms of a variable u.

First, note that a natural substitution is u = 1 + 2x^2. Then du = 4x dx, which will help to replace x dx. We also express x^2 in terms of u: x^2 = (u - 1)/2, so x^3 dx = x^2 · x dx = ((u - 1)/2) · (du/4) = (u - 1) du / 8.

Next, adjust the limits: when x = 0, u = 1 + 2(0)^2 = 1; when x = 2, u = 1 + 2(2)^2 = 9. Thus the integral becomes ∫_{u=1}^{9} [(u - 1)/8] · [1/√u] du, i.e. (1/8) ∫_{1}^{9} (u - 1)/√u du.

Now consider the given option. The expression provided in option b translates to (1/8) ∫_{1}^{9} (u - 1)/√u du, which matches the derived transformed integral exactly. The factor 1/8 is consistent with the substitution steps, and the limits 1 to 9 are correct. This option thus correctly expresses the original integral in terms of u.

If there were other options, we would compare them by checking whether they correctly implement the substitution, correctly transform x^3 dx into an expression involving du, and correctly change the limits to u = 1 and u = 9. Any option that miscomputes the differential (for example, missing the factor from x dx), misidentifies x^2 in terms of u, or uses incorrect limits would be incorrect. Conversely, an option that yields (1/8) ∫_{1}^{9} (u - 1)/√u du would be the correct form.

In summary, the correct transformed integral is (1/8) ∫_{1}^{9} (u - 1)/√u du, and option b provides exactly this expression.

Using a suitable substitution, [math: ∫02x31+2x2dx] \int_{0}^{2} \frac{x^3}{\sqrt{1+2x^2}} {dx} can be expressed in terms of [math: u] as:

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