What is $f (x) = \int {(3 x + 1)}^{2} - 6 x + 1 d x$ $f(x) = int (3x+1)^2-6x+1 dx$ if $f (2) = 1$ $f(2) = 1$ ?

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What is $f (x) = \int {(3 x + 1)}^{2} - 6 x + 1 d x$ $f(x) = int (3x+1)^2-6x+1 dx$ if $f (2) = 1$ $f(2) = 1$ ?

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Answer 1 · 2018-05-17T17:28:06

$3 x^{3} + 2 x - 27$ $3x^3+2x-27$

Explanation:

$expand and simplify$ $"expand and simplify"$

$\Rightarrow \int (9 x^{2} + 6 x + 1 - 6 x + 1) d x$ $rArrint(9x^2+6x+1-6x+1)dx$

$= \int (9 x^{2} + 2) d x$ $=int(9x^2+2)dx$

$integrate each term using the power rule$ $"integrate each term using the "color(blue)"power rule"$

$∙ x \int (a x^{n}) = \frac{a}{n + 1} x^{n + 1} \to n \neq - 1$ $•color(white)(x)int(ax^n)=a/(n+1)x^(n+1)ton!=-1$

$\Rightarrow \int (9 x^{2} + 2) d x = 3 x^{3} + 2 x + c$ $rArrint(9x^2+2)dx=3x^3+2x+c$

$where c is the constant of integration$ $"where c is the constant of integration"$

$to find c use f (2) = 1$ $"to find c use "f(2)=1$

$\Rightarrow 3 {(2)}^{3} + 2 (2) + c = 1$ $rArr3(2)^3+2(2)+c=1$

$\Rightarrow c = 1 - 28 = - 27$ $rArrc=1-28=-27$

$\Rightarrow 3 x^{3} + 2 x - 27$ $rArr3x^3+2x-27$

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What is $f (x) = \int {(3 x + 1)}^{2} - 6 x + 1 d x$ $f(x) = int (3x+1)^2-6x+1 dx$ if $f (2) = 1$ $f(2) = 1$ ?

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What is f(x) = int (3x+1)^2-6x+1 dxf(x)=∫(3x+1)2−6x+1dxf(x) = int (3x+1)^2-6x+1 dx if f(2) = 1 f(2)=1f(2) = 1 ?

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What is $f (x) = \int {(3 x + 1)}^{2} - 6 x + 1 d x$ $f(x) = int (3x+1)^2-6x+1 dx$ if $f (2) = 1$ $f(2) = 1$ ?