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###### ${selected_topic_name} • Notes • Comments & Questions Find the general form of$g\$ such that $g^{\prime}(x)=4 \sin x+\frac{x^{5}+x^{2}-3 x}{\sqrt{x}}$

$g^{\prime}(x)=4 \sin x+\frac{x^{5}+x^{2}-3 x}{\sqrt{x}}$

$=4 \sin x+\left(x^{5}+x^{2}-3 x\right) x^{-1 / 2}$

$g^{\prime}(x)=4 \sin x+x^{9 / 2}+x^{3 / 2}-3 x^{1 / 2}$

$g(x)=-4 \cos x+\frac{x^{11 / 2}}{11 / 2}+\frac{x^{5 / 2}}{5 / 2}-\frac{3 x^{3 / 2}}{3 / 2}+C$

$g(x)=-4 \cos x+2 / 11 \sqrt{x^{11}}+2 / 5 \sqrt{x^{5}}-\frac{2}{1} \sqrt{x^{3}}+c$

Find $f(x)$ if $F^{\prime}(x)=\frac{1}{x^{2}+1}+e^{x}+5$

$f(x)=\tan ^{-1}(x)+e^{x}+5 x+c$

Find $F(x)$ if $\int \frac{1}{x^{2}+1}+e^{x}+5 \quad \int \frac{1}{x}+\sec ^{2} x+\csc x \cot x \quad \int \frac{x^{2}-1}{x}$

$F(x)=\int \frac{1}{x^{2}+1}+e^{x}+5=\tan ^{-1}(x)+e^{x}+5 x+c$

$F(x)=\int \frac{1}{x}+\sec ^{2} x+\csc x \cot x=\ln (x)+\tan (x)+-\csc (x)+C$

$F(x)=\int \frac{x^{2}-1}{x}=$

$=\int \frac{x^{2}}{x}-\frac{1}{x}=\int x-\frac{1}{x}$

$F(x)=\frac{x^{2}}{2}-\ln (x)+C$