`f(x) = 4 - 3(1 + x^2)^-1, F(1) = 0` Find the antiderivative `F` that satisfies the given condition. Check your answer by comparing the graphs of `f` and `F`.

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`f(x)=4-3(1+x^2)^-1`

`F=int(4-3(1+x^2)^-1)dx`

`F=int4dx-int3/(1+x^2)dx`

`F=4x-3arctan(x)+c_1`

Let's find constant c_1 , given F(1)=0

`F(1)=0=4(1)-3arctan(1)+c_1`

`0=4-3(pi/4)+c_1`

`c_1=(3pi)/4-4`

`F=4x-3arctan(x)+(3pi)/4-4`

See the attached graph. Function f(x) is in red color , F is in blue color. From the graph also F(1)=0.

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`f(x)=4-3(1+x^2)^-1`

`F=int(4-3(1+x^2)^-1)dx`

`F=int4dx-int3/(1+x^2)dx`

`F=4x-3arctan(x)+c_1`

Let's find constant c_1 , given F(1)=0

`F(1)=0=4(1)-3arctan(1)+c_1`

`0=4-3(pi/4)+c_1`

`c_1=(3pi)/4-4`

`F=4x-3arctan(x)+(3pi)/4-4`

See the attached graph. Function f(x) is in red color , F is in blue color. From the graph also F(1)=0.

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