7–64. Integration review Evaluate the following integrals.
8. ∫ (9x - 2)^(-3) dx

Guida verificata passo dopo passo

Step 1: Recognize that the integral involves a power of a linear function, (9x - 2)^(-3). To solve this, use the substitution method. Let u = 9x - 2, which simplifies the expression.

Step 2: Compute the derivative of u with respect to x. Since u = 9x - 2, we find that du/dx = 9, or equivalently, dx = du/9.

Step 3: Rewrite the integral in terms of u. Substituting u and dx into the integral, it becomes ∫ u^(-3) * (1/9) du.

Step 4: Factor out the constant 1/9 from the integral. The integral now simplifies to (1/9) ∫ u^(-3) du.

Step 5: Apply the power rule for integration. Recall that ∫ u^n du = (u^(n+1))/(n+1) for n ≠ -1. Here, n = -3, so the integral becomes (1/9) * (u^(-3+1))/(-3+1). Simplify the expression and substitute back u = 9x - 2 to express the result in terms of x.

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