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

The integral of xsin(x^2)cos(x^2) is -1/8 cos(2x^2)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

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

- \frac{1}{8} cos (2 x^{2}) + C

Solution steps

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

Rewrite using trig identities

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

Apply u-substitution

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

= \frac{1}{2} \cdot \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}{2} \cdot \frac{1}{4} \cdot \int sin (u) d u

Use the common integral:

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

= \frac{1}{2} \cdot \frac{1}{4} (- cos (u))

Substitute back

u = 2 x^{2}

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

Simplify

\frac{1}{2} \cdot \frac{1}{4} (- cos (2 x^{2})) : - \frac{1}{8} cos (2 x^{2})

= - \frac{1}{8} cos (2 x^{2})

Add a constant to the solution

= - \frac{1}{8} cos (2 x^{2}) + C

\int cos^{7} (x) sin^{5} (x) d x

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

\int e^{t} + 1 d t

\int \frac{1 + cos ^{2} ( θ )}{cos ^{2} ( θ )} d θ

\int \frac{1}{x ( 4 + x ^{2} ) ^{3}} d x

What is the integral of xsin(x^2)cos(x^2) ?
The integral of xsin(x^2)cos(x^2) is -1/8 cos(2x^2)+C