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

The integral from 0 to 1 of sin(pix^2)x is 1/pi

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integral from 0 to 1 of sin(pix^2)x

\int_{0}^{1} sin (π x^{2}) x d x

\frac{1}{π}

Decimal Notation

0.31830 \dots

Solution steps

\int_{0}^{1} sin (π x^{2}) x d x

Apply u-substitution

: \int_{0}^{π} \frac{sin ( u )}{2 π} d u

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Use the common integral:

\int sin (u) d u = - cos (u)

= \frac{1}{2 π} [- cos (u)]_{0}^{π}

Compute the boundaries:

2

= \frac{1}{2 π} \cdot 2

Simplify

= \frac{1}{π}

\int_{0}^{300} (1000 - 5 x - 140) d x

\int_{0}^{2} x 1 + x d x

\int_{0}^{\frac{π}{2}} \frac{1}{2}

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

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

What is the integral from 0 to 1 of sin(pix^2)x ?
The integral from 0 to 1 of sin(pix^2)x is 1/pi