What is the integral of (4^x)/(sqrt(2))-2sqrt(x) ?

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

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integral of (4^x)/(sqrt(2))-2sqrt(x)

\int \frac{4 ^{x}}{2} - 2 x d x

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

Solution steps

\int \frac{4 ^{x}}{2} - 2 x 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 \frac{4 ^{x}}{2} d x - \int 2 x d x

\int \frac{4 ^{x}}{2} d x = \frac{2 ^{\frac{1}{2} (4 x - 3)}}{ln ( 2 )}

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

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

Add a constant to the solution

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

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