Integrate ∫
1
0
3
√
1+7x
dx .

Question

BlackTom AI · Accepted Answer

The problem asks to evaluate the definite integral ∫ from 0 to 1 of ∛(1 + 7x) dx, where ∛ denotes the cube root. To handle this, I’ll perform a substitution to simplify the exponent.

First, set u = 1 + 7x. Then du = 7 dx, so dx = du/7. When x = 0, u = 1; when x = 1, u = 8. The integral becomes (1/7) ∫ from 1 to 8 of u^(1/3) du.

Integrating u^(1/3) with respect to u gives (u^(4/3)) / (4/3) = (3/4) u^(4/3). Including the 1/7 factor, the result is (3/28) [u^(4/3)] from 1 to 8.

Now compute the boundary values: 8^(4/3) = (8^(1/3))^4 = 2^4 = 16, and 1^(4/3) = 1. Therefore the definite integral equals (3/28) (16 − 1) = (3/28) × 15 = 45/28.

Option-by-option analysis:
- 45/28: This matches the computed value exactly, so it is the correct result.
- 45/4: This would imply a much larger area; it would correspond to (45/4) ≈ 11.25, which is inconsistent with the small interval [0,1] and the cube-root growth.
- 4/21: This is a very small value (~0.190), far from the actual ~1.607, so it underestimates the integral.
- 15/7: This is ≈ 2.142, also too large for the same reason; the average value of ∛(1+7x) over [0,1] cannot yield such a big product with length 1.
- 3/4: This is 0.75, which is again not matching the actual area; the actual integral is greater because the function increases from ∛1 = 1 to ∛8 = 2, averaging near 1.6.
- 15: This would be far too large; it would require a much larger area than the interval provides.
- 45/21: This simplifies to 15/7 ≈ 2.143, which is not consistent with the computed 45/28 ≈ 1.607.
- 3/28: This is about 0.107, which is far smaller than the actual integral value.

In summary, the correct evaluation is 45/28, and the other options do not align with the definite integral’s value given the substitution and bounds.

MATH-112-301-001 Proctored Final Exam

Integrate ∫ 1 0 3 √ 1+7x dx .

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