Question

In Exercises \(17-24,\) use the indicated substitution to evaluate the integral. Confirm your answer by differentiation. $$\int 28(7 x-2)^{3} d x, \quad u=7 x-2$$

Short answer

The integral of \(28(7x - 2)^3\) with respect to \(x\), using substitution \(u = 7x - 2\), is \((7x - 2)^4 + C\). Confirming the answer by differentiation we get back the original integrand.

Step-by-step solution

Perform the Substitution

Substitute \(u = 7x - 2\) in the integral. To do this, first solve for \(x\) in the substitution equation. From \(u = 7x - 2\) we get \(x = (u + 2) / 7\). Also, we need to change \(dx\) into terms of \(du\). Differentiate \(u = 7x - 2\) w.r.t \(x\) to get \(du = 7dx\). Then, solve for \(dx\), which is \(dx = du / 7\). The integral becomes \(\int 28u^3 (du / 7)\).

Simplify the Integral

Now simplify the integral by combining like terms, \(\int 28u^3 (du / 7) = 4 \int u^3 du\).

Evaluate the Integral

Evaluate the integral using power rule, which states that the integral of \(x^n = x^{n+1} / (n + 1)\). So, \(\int u^3 du = u^{3+1} / (3 + 1) = u^4 / 4 + C\), where \(C\) is the constant of integration.

Substitute Back the Original Variable

Now, substitute back \(u = 7x - 2\) into the solution, which gives us \(u^4 / 4 + C = (4(u^4) + C) = 4(7x - 2)^4 / 4 + C\). Simplify to get the final form of the integral, which is \((7x - 2)^4 + C\).

Confirm the Answer by Differentiation

Confirm that the answer is correct by differentiating the obtained result: \(d/dx [(7x - 2)^4 + C] = 4 * (7x - 2)^3 * 7 = 28 * (7x - 2)^3\), which matches the original integrand, confirming correctness.