Use a u-substitution to completely rewrite the definite integral in terms of u. Give exact values only. No decimals. ∫[8 5]10x-4/(5x^2-4x+9)^15dx=∫[b a]gud(u).

Question

u=?

du=? dx

a=?

b=?

g(u)=?

Answer 1

Hi Ryan,

In this question, we can see that the term in the numerator (10x - 4) looks like the derivative of the argument in our denominator (5x² - 4x + 9). This should give us a hint as to what our u and du should be. In this case..

u = 5x² - 4x + 9

du = (10x - 4)dx

g(u) = 1 / u¹⁵

To find the bounds with our new integral, we can simply plug in the upper and lower values of x into our u (first equation).

b = u(8) = 5(8)² - 4(8) + 9 = 297

a = u(5) = 5(5)² - 4(5) + 9 = 114

If we rewrite our integral now, by subbing u and du and our new bounds we would get...

∫[297 114] 1 / u¹⁵ du

Answer 2

∫⁸₅(10x-4)/(5x²-4x+9)¹⁵dx

u = 5x²-4x+9

du = (10x-4)dx

when x = 5, u = 114 = a

when x = 8, u =305 = b

g(u) = 1/u¹⁵

∫³⁰⁵₁₁₄1/u¹⁵du = -u^-14/14 |₁₁₄³⁰⁵ = (1/14)(1/114¹⁴ - 1/305¹⁴)

Answer 3

I hope this helps!

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