What is the integral from 0 to pi/4 of tan(x)+sin(x) ?

The integral from 0 to pi/4 of tan(x)+sin(x) is (ln(2)-sqrt(2))/2+1

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integral from 0 to pi/4 of tan(x)+sin(x)

\int_{0}^{\frac{π}{4}} tan (x) + sin (x) d x

\frac{ln ( 2 ) - 2}{2} + 1

Decimal

0.63946 \dots

Solution steps

\int_{0}^{\frac{π}{4}} tan (x) + sin (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{π}{4}} tan (x) d x + \int_{0}^{\frac{π}{4}} sin (x) d x

\int_{0}^{\frac{π}{4}} tan (x) d x = \frac{1}{2} ln (2)

\int_{0}^{\frac{π}{4}} sin (x) d x = - \frac{1}{2} + 1

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

Simplify

\frac{1}{2} ln (2) - \frac{1}{2} + 1 : \frac{ln ( 2 ) - 2}{2} + 1

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

\int_{0}^{\frac{π}{4}} cos (2 t) d t

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\int_{0}^{2} \frac{1}{3 x + 4} d x

What is the integral from 0 to pi/4 of tan(x)+sin(x) ?
The integral from 0 to pi/4 of tan(x)+sin(x) is (ln(2)-sqrt(2))/2+1