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Integrate the function f(x) = 3x2 − 4x + 2.

Mathematics

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Answer

x3−2x2+2x+Cx^3 - 2x^2 + 2x + C

Graph of f(x) = 3x² − 4x + 2

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Key points of f(x) = 3x² − 4x + 2
Pointxy
y-intercept02
Minimum0.670.67

Step-by-step solution

2 steps

  1. Apply the power rule for integration to each term.

    The power rule states that ∫xn dx=xn+1n+1+C\int x^n \,dx = \frac{x^{n+1}}{n+1} + C.

    ∫(3x2−4x+2) dx\int (3x^2 - 4x + 2) \,dx

    Integrate 3x23x^2:

    ∫3x2 dx=3x2+12+1=3x33=x3\int 3x^2 \,dx = 3 \frac{x^{2+1}}{2+1} = 3 \frac{x^3}{3} = x^3

    Integrate −4x-4x:

    ∫−4x dx=−4x1+11+1=−4x22=−2x2\int -4x \,dx = -4 \frac{x^{1+1}}{1+1} = -4 \frac{x^2}{2} = -2x^2

    Integrate 2:

    ∫2 dx=2x\int 2 \,dx = 2x

  2. Combine the integrated terms and add the constant of integration.

    x3−2x2+2x+Cx^3 - 2x^2 + 2x + C
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