Math · Calculus I · Worked example

Find an indefinite integral term by term

Find ∫(6x² − 4x + 5) dx and check the answer by differentiating.

\int (6 x^{2} - 4 x + 5) d x

Split into terms

Integrate term by term and keep each constant factor in front.

6 \int x^{2} d x - 4 \int x d x + \int 5 d x

Run the power rule backward

x² becomes x³/3 and x becomes x²/2; the constant 5 becomes 5x. One + C covers all three terms, because a sum of constants is a constant.

6 \cdot \frac{x^{3}}{3} - 4 \cdot \frac{x^{2}}{2} + 5 x + C

Simplify

6·x³/3 = 2x³ and 4·x²/2 = 2x².

2 x^{3} - 2 x^{2} + 5 x + C

Check by differentiating

The derivative of 2x³ − 2x² + 5x + C is 6x² − 4x + 5, the integrand, so the answer is right. The constant differentiates to zero, which is why any C works.

\frac{d}{d x} (2 x^{3} - 2 x^{2} + 5 x + C) = 6 x^{2} - 4 x + 5

\begin{array}{c} \frac{d}{d x} (2 x^{3} - 2 x^{2} + 5 x + C) \\ = 6 x^{2} - 4 x + 5 \end{array}

Result

∫(6x² − 4x + 5) dx = 2x³ − 2x² + 5x + C.

Your turn

Find ∫(8x³ + 3√x) dx.

Show the answer and explanation

2x⁴ + 2x^(3/2) + C.

Write √x as x^(1/2). Then 8x³ gives 8·x⁴/4 = 2x⁴, and 3x^(1/2) gives 3·x^(3/2)/(3/2) = 2x^(3/2). Differentiating returns 8x³ + 3√x.

\begin{array}{l} F (x) = 2 x^{4} + 2 x^{3 / 2} \\ F^{'} (x) = 8 x^{3} + 3 \sqrt{x} \end{array}

Keep exploring

In the checker, change the proposed antiderivative to 2x³ − 4x² + 5x + C, as if the 2 had not been divided out. Its derivative is 6x² − 8x + 5, so the check fails.

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OpenStax Calculus Volume 1 — Antiderivatives

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Verify it in the antiderivative checker Check the antiderivative in Math Open worked example on a board Antiderivative power rule in Math Reference

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