What is the integral from 0 to 1 of sin(pit+pi/4) ?

The integral from 0 to 1 of sin(pit+pi/4) is (sqrt(2))/pi

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integral from 0 to 1 of sin(pit+pi/4)

\int_{0}^{1} sin (π t + \frac{π}{4}) d t

\frac{2}{π}

Decimal

0.45015 \dots

Solution steps

\int_{0}^{1} sin (π t + \frac{π}{4}) d t

Rewrite using trig identities

= \int_{0}^{1} \frac{1}{2} cos (π t) + \frac{1}{2} sin (π t) d t

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}^{1} \frac{1}{2} cos (π t) d t + \int_{0}^{1} \frac{1}{2} sin (π t) d t

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

\int_{0}^{1} \frac{1}{2} sin (π t) d t = \frac{2}{π}

= 0 + \frac{2}{π}

Simplify

= \frac{2}{π}

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