Evaluate the integral: integral (4x2 + 4x + 4) dx

May 22, 2026Mathematics

ScanSolve AI Answer

May 22, 2026

Answer

$\frac{4}{3}x^3 + 2x^2 + 4x + C$

Here's how to solve the integral:

Step 1: Apply the sum rule for integrals. $\int (4x^2 + 4x + 4) dx = \int 4x^2 dx + \int 4x dx + \int 4 dx$

Step 2: Apply the constant multiple rule and the power rule for integration ( $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ ). $= 4 \int x^2 dx + 4 \int x^1 dx + 4 \int x^0 dx$ $= 4 \left(\frac{x^{2+1}}{2+1}\right) + 4 \left(\frac{x^{1+1}}{1+1}\right) + 4 \left(\frac{x^{0+1}}{0+1}\right) + C$ $= 4 \left(\frac{x^3}{3}\right) + 4 \left(\frac{x^2}{2}\right) + 4 \left(\frac{x^1}{1}\right) + C$

Step 3: Simplify the expression. $= \frac{4}{3}x^3 + 2x^2 + 4x + C$

The final answer is $\boxed{\frac{4}{3}x^3 + 2x^2 + 4x + C}$ .

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Quick Answer

Here's how to solve the integral: Step 1: Apply the sum rule for integrals.
(4x^2 + 4x + 4) dx = 4x^2 dx + 4x dx + 4 dx Step 2: Apply the constant multiple rule and the power rule for integration ( x^n dx = x^n+1n+1 + C).
= (4)/(3)x^3 + 2x^2 + 4x + C The final answer is (4)/(3)x^3 + 2x^2 + 4x + C.
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Asked on May 22, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

ScanSolve AI AnswerAnswered on May 22, 2026

Here's how to solve the integral: Step 1: Apply the sum rule for integrals. (4x^2 + 4x + 4) dx = 4x^2 dx + 4x dx + 4 dx Step 2: Apply the constant multiple rule and the power rule for integration ( x^n dx = x^n+1n+1 + C). = 4 x^2 dx + 4 x^1 dx + 4 x^0 dx = 4 (x^2+12+1) + 4 (x^1+11+1) + 4 (x^0+10+1) + C = 4 ((x^3)/(3)) + 4 ((x^2)/(2)) + 4 ((x^1)/(1)) + C Step 3: Simplify the expression. = (4)/(3)x^3 + 2x^2 + 4x + C The final answer is (4)/(3)x^3 + 2x^2 + 4x + C. 3 done, 2 left today. You're making progress.