Calculus • Step-by-Step

Integral of (5x⁴ - 2x³ + 7) dx with Steps

Integrate the polynomial 5x⁴ - 2x³ + 7 step by step. Apply the reverse power rule to each term to find the antiderivative x⁵ - (1/2)x⁴ + 7x + C.

Problem Statement

\int (5x^4 - 2x^3 + 7) \, dx

Final Result

F(x) = x^{5}-\frac{x^{4}}{2}+7 \cdot x + C

Formula Required: Power Rule for Polynomials

5(x⁵/5) = x⁵, and -2(x⁴/4) = -(1/2)x⁴, and 7 integrates to 7x.

\int (5x^4 - 2x^3 + 7) \, dx = x^5 - \frac{1}{2}x^4 + 7x + C

6 Steps

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

\int \left( 5 \cdot x^{4}-2 \cdot x^{3}+7 \right) \, dx

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

\int \left( 5 \cdot x^{4}-2 \cdot x^{3}+7 \right) \, dx = \int 5 \cdot x^{4} \, dx - \int 2 \cdot x^{3} \, dx + \int 7 \, dx

Integrate term $5 \cdot x^{4}$ using the Power Rule for integration.

\int 5 \cdot x^{4} \, dx = x^{5}

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

\int 2 \cdot x^{3} \, dx = \frac{x^{4}}{2}

Integrate constant $7$ to obtain $7 \cdot x$.

\int 7 \, dx = 7 \cdot x

Combine all term antiderivatives.

F(x) = x^{5}-\frac{x^{4}}{2}+7 \cdot x

Interactive Calculator

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

5(x⁵/5) - 2(x⁴/4) + 7x + C = x⁵ - (1/2)x⁴ + 7x + C.