What is the integral from 0 to 1/2 of (cos(pix)-2^x) ?

The integral from 0 to 1/2 of (cos(pix)-2^x) is 1/pi-(sqrt(2)-1)/(ln(2))

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integral from 0 to 1/2 of (cos(pix)-2^x)

\int_{0}^{\frac{1}{2}} (cos (π x) - 2^{x}) d x

\frac{1}{π} - \frac{2 - 1}{ln ( 2 )}

Solution steps

\int_{0}^{\frac{1}{2}} cos (π x) - 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_{0}^{\frac{1}{2}} cos (π x) d x - \int_{0}^{\frac{1}{2}} 2^{x} d x

\int_{0}^{\frac{1}{2}} cos (π x) d x = \frac{1}{π}

\int_{0}^{\frac{1}{2}} 2^{x} d x = \frac{2 - 1}{ln ( 2 )}

= \frac{1}{π} - \frac{2 - 1}{ln ( 2 )}

\int_{0}^{1} (e^{t} - e + 1)^{2} d t

\int_{0}^{1} 1

\int_{1}^{2} (2 - 6 y^{2}) d y

\int_{2}^{5} 2 π y (5 - y) d y

\int_{0}^{4} \frac{x ^{2}}{16} d x

What is the integral from 0 to 1/2 of (cos(pix)-2^x) ?
The integral from 0 to 1/2 of (cos(pix)-2^x) is 1/pi-(sqrt(2)-1)/(ln(2))