Question

Using the substitution u=5x−2, ∫x5x−2−−−−−√ⅆx is equivalent to which of the following?

Answer 1

To evaluate the integral ∫(√x)/(5x-2) dx using the substitution u = 5x - 2, follow these steps: differentiate u with respect to x, rearrange the equation to solve for dx, substitute u and dx into the integral expression, and finally integrate the expression with respect to u. The result is (2/3)(√u)² - 2√u + C.

Step-by-step explanation:

To evaluate the integral ∫(√x)/(5x-2) dx using the substitution u = 5x - 2, follow these steps:

Differentiate u with respect to x: du/dx = 5.

Rearrange the equation to solve for dx: dx = du/5.

Substitute u and dx into the integral expression: ∫(√x)/(5x-2) dx = ∫(√u)/(5(u + 2)/5) ⋅ (1/5) du = ∫(√u)/(u + 2) du.

Now integrate the expression with respect to u: ∫(√u)/(u + 2) du = (2/3)(√u)² - 2√u + C, where C is the constant of integration.

Therefore, the integral ∫(√x)/(5x-2) dx is equivalent to (2/3)(√u)² - 2√u + C.

Answer 2

To evaluate the given integral using the substitution u=5x-2, follow the steps of differentiation, substitution, and evaluation.

Step-by-step explanation:

To evaluate the integral ∫(x⁵)/(√(5x-2))dx using the substitution u=5x-2, we need to follow these steps:

1. Find du/dx by differentiating the expression u=5x-2. We get du/dx = 5.
2. Replace dx with du/5 in the original integral, and replace x with (u+2)/5.
3. Simplify the integrand using this substitution, and change the limits of integration if necessary.
4. Evaluate the integral using the new limits and simplify the result.

Overall, the integral using the substitution u=5x-2 will be equivalent to another integral with different limits, but the integrand will be simplified.