What is the solution for f^'(x)=12x^2-6x+1,f(1)=5 ?

The solution for f^'(x)=12x^2-6x+1,f(1)=5 is f(x)=4x^3-3x^2+x+3

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f^'(x)=12x^2-6x+1,f(1)=5

f^{^{'}} (x) = 12 x^{2} - 6 x + 1, f (1) = 5

f (x) = 4 x^{3} - 3 x^{2} + x + 3

Solution steps

f^{'} (x) = 12 x^{2} - 6 x + 1

f^{^{'}} (x) = g (x)

then

f (x) = \int g (x) d x

f (x) = \int 12 x^{2} - 6 x + 1 d x

\int 12 x^{2} - 6 x + 1 d x = 4 x^{3} - 3 x^{2} + x + c_{1}

f (x) = 4 x^{3} - 3 x^{2} + x + c_{1}

Apply initial conditions:

f (x) = 4 x^{3} - 3 x^{2} + x + 3

f (x) = 4 x^{3} - 3 x^{2} + x + 3

\int \frac{5}{x ( 1 + x )} d x

\frac{1}{2}

x^{2} 1 + x^{2}

\int (- x^{2} - 3 x + 3)^{2} d x

\frac{d}{d x} (- \frac{1}{6} x^{3} - \frac{1}{2} x^{2} + x + 1 + k e^{k})

What is the solution for f^'(x)=12x^2-6x+1,f(1)=5 ?
The solution for f^'(x)=12x^2-6x+1,f(1)=5 is f(x)=4x^3-3x^2+x+3