Question

The following integral may be solved using substitution. choose a function u that may be used to solve the problem. then find du.

a) ∫(5x⁴ - 4) * 4x² dx
b) ∫ √4-x dx
c) ∫ x⁴/3x⁵ + 4
d) ∫ 6x⁵ eˣ⁶ dx

(a) Let u=_ so that du=dx.
(b) Let u =_ so that du= dx
(c) Let u = _ so that du = dx
(d) Let us =_ so that du=dx.

Answer 1

To apply substitution to the given integrals, we choose a substitution function u for each based on what simplifies the integrals. The formulas for u and du are given for each integral, allowing them to be recalculated in terms of u, making the integration process simpler.

Step-by-step explanation:

To solve the given integrals using substitution, we choose a function u for each integral and then find du. For each integral, we select u such that the derivative du will simplify the integration process. Here are the substitutions for the given integrals:

• (a) Let u = 5x^4 - 4 so that du = 20x^3 dx. Since there is a 4x^2 dx present, we can adjust du to match by dividing by 5: du/5 = 4x^3 dx. Then perform the substitution in the integral.
• (b) For u = 4 - x, it follows that du = -dx, or equivalently dx = -du. The integral becomes an integral in terms of u with the square root of u.
• (c) In this integral, u = 3x^5 + 4 is a suitable substitution since the numerator is a derivative of u. So, du = 15x^4 dx. We can adjust for the x^4 in the integral by dividing du by 15.
• (d) Finally, u = x^6 is chosen for this integral because the exponential function's argument is x^6. This gives us du = 6x^5 dx, which matches the 6x^5 in the integral exactly.

By applying these substitutions, we transform each integral into a form that is easier to solve.