Evaluate the integral: integral (3x−2)5 dx

Question

ScanSolve AI Tutor · Accepted Answer

To solve the integral ∫ (3x−2)⁵ dx, we can use the substitution method. Step 1: Set up the substitution. Let u = 3x − 2. Step 2: Differentiate u with respect to x to find du. du/dx = d/dx(3x − 2) = 3; This means du = 3 dx. Step 3: Express dx in terms of du. dx = 1/3 du Step 4: Substitute u and dx into the original integral. ∫ (3x−2)⁵ dx = ∫ u⁵ (1/3 du); = 1/3 ∫ u⁵ du; Step 5: Integrate with respect to u using the power rule for integration, ∫ uⁿ du = (uⁿ⁺¹)/(n+1) + C. 1/3 ∫ u⁵ du = 1/3 · (u⁵⁺¹)/(5+1) + C; = 1/3 · (u⁶)/(6) + C; = (u⁶)/(18) + C; Step 6: Substitute back u = 3x − 2 to express the answer in terms of x. ((3x−2)⁶)/(18) + C The final answer is ((3x−2)⁶)/(18) + C.

Evaluate the integral: integral (3x-2)5 dx

Answer

Step-by-step solution

Set up the substitution.

Differentiate $u$ with respect to $x$ to find $du$ .

Express $dx$ in terms of $du$ .

Substitute $u$ and $dx$ into the original integral.

Integrate with respect to $u$ using the power rule for integration, $\int u^n du = \frac{u^{n+1}}{n+1} + C$ .

Substitute back $u = 3x - 2$ to express the answer in terms of $x$ .

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