IntegralsEasy

Indefinite integral — power and logarithm

Calculate the indefinite integral ∫ (3x² + 2/x) dx.

Given data

∫ (3x² + 2/x) dx

Steps

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Integrate term by term: ∫ 3x² dx = ?
Write your working — the numerical result must appear in the text (logica)
Integrate ∫ (2/x) dx = ?
Combine the results. What is the total integral?

Full worked solution

Integrate term by term: ∫ 3x² dx = ?
$\int 3x^2\,dx = 3 \cdot \dfrac{x^3}{3} = x^3$
$\int 3x^2\,dx = 3 \cdot \frac{x^{3}}{3} + C_1 = \mathbf{x^3} + C_1$ . Applying the power rule $\int x^n dx = x^{n+1}/(n+1)$ with $n = 2$ .
Integrate ∫ (2/x) dx = ?
$\int \dfrac{2}{x}\,dx = 2\ln|x|$
$\int \frac{2}{x}\,dx = 2\ln|x| + C_2$ . The integral of $1/x$ is $\ln|x|$ , a standard result.
Combine the results. What is the total integral?
$\int (3x^2 + 2/x)\,dx = x^3 + 2\ln|x| + C$
$\int (3x^2 + 2/x)\,dx = x^3 + 2\ln|x| + \mathbf{C}$ . Combining the two terms and adding a single arbitrary constant $C = C_1 + C_2$ .

Result:x³ + 2 ln|x| + C.

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