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

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

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

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

\frac{2}{π}

Decimal

0.45015 \dots

Solution steps

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

Rewrite using trig identities

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

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

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

= 0 + \frac{2}{π}

Simplify

= \frac{2}{π}

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