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

The integral of x*sin(2x^2) is -1/4 cos(2x^2)+C

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integral of x*sin(2x^2)

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

- \frac{1}{4} cos (2 x^{2}) + C

Solution steps

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

Apply u-substitution

: \int \frac{sin ( u )}{4} d u

= \int \frac{sin ( u )}{4} d u

Take the constant out:

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

= \frac{1}{4} \cdot \int sin (u) d u

Use the common integral:

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

= \frac{1}{4} (- cos (u))

Substitute back

u = 2 x^{2}

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

Simplify

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

Add a constant to the solution

= - \frac{1}{4} cos (2 x^{2}) + C

x \to 0 + lim (x^{2} e^{\frac{1}{x}})

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

\frac{d}{d x} (2 e^{- 2 x} + \frac{1}{3} e^{x})

\frac{d}{d x} (6^{x^{2} - 3})

\int_{0}^{\frac{π}{2}} sin^{3} (\frac{x}{2}) d x

What is the integral of x*sin(2x^2) ?
The integral of x*sin(2x^2) is -1/4 cos(2x^2)+C