Calculus • Step-by-Step

Integral of (4x³ - 3x² + 2x - 1) dx with Steps

Step-by-step antiderivative of a cubic polynomial. Term-by-term integration produces x⁴ - x³ + x² - x + C.

Problem Statement

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

Final Result

F(x) = x^{4}-x^{3}+x^{2}-x + C

Formula Required: Term-by-Term Polynomial Integration

Integrate each power term using the power rule.

\int \sum c_i x^i \, dx = \sum \frac{c_i x^{i+1}}{i+1} + C

7 Steps

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

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

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

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

Integrate term $4 \cdot x^{3}$ using the Power Rule for integration.

\int 4 \cdot x^{3} \, dx = x^{4}

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

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

Integrate term $2 \cdot x$ using the Power Rule for integration.

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

Integrate constant $1$ to obtain $x$.

\int 1 \, dx = x

Combine all term antiderivatives.

F(x) = x^{4}-x^{3}+x^{2}-x

Interactive Calculator

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

Integrate term by term: 4(x⁴/4) - 3(x³/3) + 2(x²/2) - x + C = x⁴ - x³ + x² - x + C.