What is the integral of (4e^x+12x^{3/2}+4x^{1/2}) ?

The integral of (4e^x+12x^{3/2}+4x^{1/2}) is 4e^x+24/5 x^{5/2}+8/3 x^{3/2}+C

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integral of (4e^x+12x^{3/2}+4x^{1/2})

\int (4 e^{x} + 12 x^{\frac{3}{2}} + 4 x^{\frac{1}{2}}) d x

4 e^{x} + \frac{24}{5} x^{\frac{5}{2}} + \frac{8}{3} x^{\frac{3}{2}} + C

Solution steps

\int 4 e^{x} + 12 x^{\frac{3}{2}} + 4 x^{\frac{1}{2}} d x

Apply the Sum Rule:

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

= \int 4 e^{x} d x + \int 12 x^{\frac{3}{2}} d x + \int 4 x^{\frac{1}{2}} d x

\int 4 e^{x} d x = 4 e^{x}

\int 12 x^{\frac{3}{2}} d x = \frac{24}{5} x^{\frac{5}{2}}

\int 4 x^{\frac{1}{2}} d x = \frac{8}{3} x^{\frac{3}{2}}

= 4 e^{x} + \frac{24}{5} x^{\frac{5}{2}} + \frac{8}{3} x^{\frac{3}{2}}

Add a constant to the solution

= 4 e^{x} + \frac{24}{5} x^{\frac{5}{2}} + \frac{8}{3} x^{\frac{3}{2}} + C

x^{2} - 4, at x = 3

\frac{ln ( x )}{10 x}

\frac{d}{d x} (\frac{2 x + cos ( 2 x )}{x ^{2} + sin ( 2 x )})

\frac{\partial}{\partial x} ((x^{2} + y^{2}) e^{- x})

x \to 1 lim (\frac{4}{x ^{2} - 1})

What is the integral of (4e^x+12x^{3/2}+4x^{1/2}) ?
The integral of (4e^x+12x^{3/2}+4x^{1/2}) is 4e^x+24/5 x^{5/2}+8/3 x^{3/2}+C