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integral from 0 to x of sin(t^2)cos(t^2)

Solution

\int_{0}^{x} sin (t^{2}) cos (t^{2}) d t

\frac{1}{2 2} (\frac{π}{2} § (\frac{2}{π} 2 x) - \frac{π}{2} § (\frac{2}{π} \cdot 0))

Solution steps

\int_{0}^{x} sin (t^{2}) cos (t^{2}) d t

Rewrite using trig identities

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

Take the constant out:

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

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

Apply Integral Substitution

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

Take the constant out:

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

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

Apply exponent rule:

\frac{a ^{n}}{b ^{n}} = (\frac{a}{b})^{n}

\frac{u ^{2}}{2} = (\frac{u}{2})^{2}

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

Apply u-substitution

: \int_{0}^{2 x} sin (v^{2}) 2 d v

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

Take the constant out:

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

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

Use the nonelementary integral:

\int sin (x^{2}) d x = \frac{π}{2} § (\frac{2}{π} x)

= \frac{1}{2} \cdot \frac{1}{2} 2 [\frac{π}{2} § (\frac{2}{π} v)]_{0}^{2 x}

Simplify

\frac{1}{2} \cdot \frac{1}{2} 2 [\frac{π}{2} § (\frac{2}{π} v)]_{0}^{2 x} : \frac{1}{2 2} [\frac{π}{2} § (\frac{2}{π} v)]_{0}^{2 x}

= \frac{1}{2 2} [\frac{π}{2} § (\frac{2}{π} v)]_{0}^{2 x}

Compute the boundaries:

\frac{π}{2} § (\frac{2}{π} 2 x) - \frac{π}{2} § (\frac{2}{π} \cdot 0)

= \frac{1}{2 2} (\frac{π}{2} § (\frac{2}{π} 2 x) - \frac{π}{2} § (\frac{2}{π} \cdot 0))

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