Calculus • Step-by-Step

Integral of (3x² - 4x + 1) dx with Steps

Integrate the quadratic polynomial 3x² - 4x + 1 term by term with complete working to obtain x³ - 2x² + x + C.

Problem Statement

\int (3x^2 - 4x + 1) \, dx

Final Result

F(x) = x^{3}-2 \cdot x^{2}+x + C

Formula Required: Polynomial Integration

Integrate each power of x using the power rule.

\int (ax^2 + bx + c) \, dx = \frac{a x^3}{3} + \frac{b x^2}{2} + cx + C

6 Steps

Set up the indefinite integral with respect to variable $x$.

\int \left( 3 \cdot x^{2}-4 \cdot x+1 \right) \, dx

Apply the Sum & Difference Rule: integrate each term separately.

\int \left( 3 \cdot x^{2}-4 \cdot x+1 \right) \, dx = \int 3 \cdot x^{2} \, dx - \int 4 \cdot x \, dx + \int 1 \, dx

Integrate term $3 \cdot x^{2}$ using the Power Rule for integration.

\int 3 \cdot x^{2} \, dx = x^{3}

Integrate term $4 \cdot x$ using the Power Rule for integration.

\int 4 \cdot x \, dx = 2 \cdot x^{2}

Integrate constant $1$ to obtain $x$.

\int 1 \, dx = x

Combine all term antiderivatives.

F(x) = x^{3}-2 \cdot x^{2}+x

Interactive Calculator

Enter any equation like 2x + 8 = 20 or x² - 5x + 6 = 0 to get step-by-step solutions.

3(x³/3) - 4(x²/2) + 1x + C = x³ - 2x² + x + C.