Problem 46

Evaluate the definite integral. $$ \int_{2}^{4} \frac{x^{2}}{(3 x-5)} d x $$

Verified

Answer

The exact value of the integral is $ 44 $.

1Step 1: Set the Value of U

Set the substitution variable, $ u = 3x - 5 $, which simplifies the denominator.

2Step 2: Find the value of dx

Differentiate $ u = 3x - 5 $ with respect to $ x $ to find $ du/dx = 3 $, from which we get $ dx = du/3 $.

3Step 3: Substitute in the Integral

Substitute $ u $ and $ dx $ in the integral to get a new integral: $ \int_{1}^{3} \frac{(u + 5)^2}{u} * (du/3) $.

4Step 4: Simplify the Integral

Simplify the integrand to get $ \int_{1}^{3} (u/3 + 5)^2 du $.

5Step 5: Evaluate the Integral

Evaluate the integral to obtain $ \frac{1}{9}u^3 + 5u^2 + 25u |_{1}^{3} $.

6Step 6: Substitute U Back

Substitute $ u = 3x - 5 $ back in to get $ \frac{1}{9}(3x - 5)^3 + 5x^2 + 25x $ and evaluate this from 2 to 4.

7Step 7: Final Answer

Calculate the final answer by using the fundamental theorem of calculus: $ F(4) - F(2) $ where $ F(x) = \frac{1}{9}(3x - 5)^3 + 5x^2 + 25x $.