What is the integral of cos(x^2)sin(x^2) ?

The integral of cos(x^2)sin(x^2) is (sqrt(pi))/4 \S(sqrt(2/pi)(2x)/(sqrt(2)))+C

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integral of cos(x^2)sin(x^2)

\int cos (x^{2}) sin (x^{2}) d x

\frac{π}{4} § (\frac{2}{π} \frac{2 x}{2}) + C

Solution steps

\int cos (x^{2}) sin (x^{2}) d x

Rewrite using trig identities

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

Take the constant out:

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

= \frac{1}{2} \cdot \int sin (2 x^{2}) d x

Apply Integral Substitution

= \frac{1}{2} \cdot \int \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 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 sin ((\frac{u}{2})^{2}) d u

Apply u-substitution

: \int sin (v^{2}) 2 d v

= \frac{1}{2} \cdot \frac{1}{2} \cdot \int 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 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)

Substitute back

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

Simplify

\frac{1}{2} \cdot \frac{1}{2} 2 \frac{π}{2} § (\frac{2}{π} \frac{2 x}{2}) : \frac{π}{4} § (\frac{2}{π} \frac{2 x}{2})

= \frac{π}{4} § (\frac{2}{π} \frac{2 x}{2})

Add a constant to the solution

= \frac{π}{4} § (\frac{2}{π} \frac{2 x}{2}) + C

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What is the integral of cos(x^2)sin(x^2) ?
The integral of cos(x^2)sin(x^2) is (sqrt(pi))/4 \S(sqrt(2/pi)(2x)/(sqrt(2)))+C