What is the integral of (5^x-1/(2\sqrt[3]{x)}+1/2) ?

The integral of (5^x-1/(2\sqrt[3]{x)}+1/2) is (5^x)/(ln(5))-3/4 x^{2/3}+1/2 x+C

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integral of (5^x-1/(2\sqrt[3]{x)}+1/2)

\int (5^{x} - \frac{1}{2 3 x} + \frac{1}{2}) d x

\frac{5 ^{x}}{ln ( 5 )} - \frac{3}{4} x^{\frac{2}{3}} + \frac{1}{2} x + C

Solution steps

\int 5^{x} - \frac{1}{2 3 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 5^{x} d x - \int \frac{1}{2 3 x} d x + \int \frac{1}{2} d x

\int 5^{x} d x = \frac{5 ^{x}}{ln ( 5 )}

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

\int \frac{1}{2} d x = \frac{1}{2} x

= \frac{5 ^{x}}{ln ( 5 )} - \frac{3}{4} x^{\frac{2}{3}} + \frac{1}{2} x

Add a constant to the solution

= \frac{5 ^{x}}{ln ( 5 )} - \frac{3}{4} x^{\frac{2}{3}} + \frac{1}{2} x + C

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What is the integral of (5^x-1/(2\sqrt[3]{x)}+1/2) ?
The integral of (5^x-1/(2\sqrt[3]{x)}+1/2) is (5^x)/(ln(5))-3/4 x^{2/3}+1/2 x+C