Calculus AAnti- Derivatives And Indefinite Integral Problems 1

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 $$\int \frac{1}{x^{2}+1}+e^{x}+5 \quad \int \frac{1}{x}+\sec ^{2} x+\csc x \cot x \quad \int \frac{x^{2}-1}{x}$$

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

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

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

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

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