What is f(x) = int (3x+5)^2-x dxf(x)=∫(3x+5)2−xdx if f(1)=2 f(1)=2?

1 Answer
mason m
Jan 28, 2016

f(x)=3x^3+29/2x^2+25x-81/2f(x)=3x3+292x2+25x−812

Explanation:

The easiest way to integrate this is by expanding (3x+5)^2(3x+5)2 and subtracting xx to get

int9x^2+29x+25dx∫9x2+29x+25dx

We can now integrate each term individually using the rule

intx^ndx=(x^(n+1))/(n+1)+C∫xndx=xn+1n+1+C

This gives us the indefinite integral of

=(9x^(2+1))/(2+1)+(29x^(1+1))/(1+1)+(25x^(0+1))/(0+1)+C=9x2+12+1+29x1+11+1+25x0+10+1+C

=3x^3+29/2x^2+25x+C=3x3+292x2+25x+C

We can now find CC, the constant of integration for this particular function, since we know that f(1)=2f(1)=2.

f(1)=3(1^3)+29/2(1^2)+25(1)+C=2f(1)=3(13)+292(12)+25(1)+C=2

85/2+C=2852+C=2

C=-81/2C=−812

Thus,

f(x)=3x^3+29/2x^2+25x-81/2f(x)=3x3+292x2+25x−812

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