Use Integration by Substitution to integrate and find the bounded area for:[math: ∫034x4x5+8dx]Write your answer correct to 4 decimal places.

Question

BlackTom AI · Accepted Answer

Begin by identifying a substitution that simplifies the integral. The integrand is 4x^4/(x^5+8) with respect to x. Notice that the denominator x^5+8 has a derivative of 5x^4, which suggests the substitution u = x^5 + 8.

With u = x^5 + 8, we have du = 5x^4 dx, so x^4 dx = du/5. The integral becomes ∫ (4x^4)/(x^5+8) dx = ∫ (4/ (u)) · (du/5) = (4/5) ∫ (1/u) du.

This yields (4/5) ln|u| + C. Re-substitute u = x^5 + 8 to obtain (4/5) ln|x^5 + 8| + C.

Now evaluate from the bounds x = 0 to x = 3:
- At x = 3: x^5 + 8 = 3^5 + 8 = 243 + 8 = 251.
- At x = 0: x^5 + 8 = 0 + 8 = 8.
Thus the definite integral is (4/5)[ln(251) − ln(8)] = (4/5) ln(251/8).

For a numerical value, compute ln(251) and ln(8): ln(251) ≈ 5.525453, ln(8) ≈ 2.079442, so the difference is ≈ 3.446011. Multiplying by 4/5 gives ≈ 2.756809. Rounding to four decimal places yields 2.7568.

If you want to cross-check, you can also evaluate the expression (4/5) ln(251/8) directly and confirm the same decimal result. The key steps are the substitution u = x^5 + 8 and careful evaluation of the logarithmic difference at the bounds.

Use Integration by Substitution to integrate and find the bounded area for:[math: ∫034x4x5+8dx]Write your answer correct to 4 decimal places.Numerical

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