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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of sin(2x^2)

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

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

Solution steps

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

Apply Integral Substitution

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

Apply u-substitution

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

= \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} 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} 2 \frac{π}{2} § (\frac{2}{π} v)

Substitute back

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

Simplify

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

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

Add a constant to the solution

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

\int (cos^{4} (x) - cos^{6} (x)) d x

\int \frac{2 + ln ( 7 x )}{3 x} d x

y^{^{'}} + 4 x^{3} y = x^{3}

\frac{\partial}{\partial x} (- 5 x y + 2)

\frac{d}{d x} (4 x^{2} - 1)

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