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

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

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integral from 0 to pi/4 of cos(x)-sin(x)

\int_{0}^{\frac{π}{4}} cos (x) - sin (x) d x

2 - 1

Decimal

0.41421 \dots

Solution steps

\int_{0}^{\frac{π}{4}} cos (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}} cos (x) d x - \int_{0}^{\frac{π}{4}} sin (x) d x

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

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

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

Simplify

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

= 2 - 1

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