Integrate (6x2 - x + 3) dx

Question

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Step 1: Apply the linearity property of integration, which states that the integral of a sum or difference is the sum or difference of the integrals, and constants can be pulled out of the integral. ∫ (6x² − x + 3) dx = ∫ 6x² dx − ∫ x dx + ∫ 3 dx; Step 2: Apply the power rule for integration, which is ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C (for n ≠ −1), to each term. For the first term, 6x²: ∫ 6x² dx = 6 · (x²⁺¹)/(2+1) = 6 · (x³)/(3) For the second term, −x (which is −x¹): ∫ −x dx = − (x¹⁺¹)/(1+1) = − (x²)/(2) For the third term, 3 (which is 3x⁰): ∫ 3 dx = 3 · (x⁰⁺¹)/(0+1) = 3 · (x¹)/(1) = 3x Step 3: Combine the results from each term and add the constant of integration, C. 6 · (x³)/(3) − (x²)/(2) + 3x + C Step 4: Simplify the expression. 2x³ − (x²)/(2) + 3x + C The final answer is 2x³ − (x²)/(2) + 3x + C.

Integrate (6x2 - x + 3) dx

Answer

Step-by-step solution

Apply the linearity property of integration, which states that the integral of a sum or difference is the sum or difference of the integrals, and constants can be pulled out of the integral.

Apply the power rule for integration, which is $\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$ (for $n \neq -1$ ), to each term.

Combine the results from each term and add the constant of integration, $C$ .

Simplify the expression.

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Integrate (6x2 - x + 3) dx

Answer

Step-by-step solution

Apply the linearity property of integration, which states that the integral of a sum or difference is the sum or difference of the integrals, and constants can be pulled out of the integral.

Apply the power rule for integration, which is ∫xn dx=xn+1n+1+C\int x^n\,dx = \frac{x^{n+1}}{n+1} + C∫xndx=n+1xn+1​+C (for n≠−1n \neq -1n=−1), to each term.

Combine the results from each term and add the constant of integration, CCC.

Simplify the expression.

Try one yourself

Still stuck on this one?

More Mathematics Questions

Apply the power rule for integration, which is $\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$ (for $n \neq -1$ ), to each term.

Combine the results from each term and add the constant of integration, $C$ .