What is the integral of (x+1)y^2sin((xy)/3) ?

The integral of (x+1)y^2sin((xy)/3) is 3(-ycos((yx)/3)-yxcos((yx)/3)+3sin((yx)/3))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of (x+1)y^2sin((xy)/3)

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

3 (- y cos (\frac{y x}{3}) - y x cos (\frac{y x}{3}) + 3 sin (\frac{y x}{3})) + C

Solution steps

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

Take the constant out:

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

= y^{2} \cdot \int (1 + x) sin (\frac{y x}{3}) d x

Apply u-substitution

= y^{2} \cdot \int \frac{3 sin ( u ) ( y + 3 u )}{y ^{2}} d u

Take the constant out:

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

= y^{2} \frac{3}{y ^{2}} \cdot \int sin (u) (y + 3 u) d u

Expand

sin (u) (y + 3 u) : y sin (u) + 3 u sin (u)

= y^{2} \frac{3}{y ^{2}} \cdot \int y sin (u) + 3 u sin (u) d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= y^{2} \frac{3}{y ^{2}} (\int y sin (u) d u + \int 3 u sin (u) d u)

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

\int 3 u sin (u) d u = 3 (- u cos (u) + sin (u))

= y^{2} \frac{3}{y ^{2}} (- y cos (u) + 3 (- u cos (u) + sin (u)))

Substitute back

u = \frac{y x}{3}

= y^{2} \frac{3}{y ^{2}} (- y cos (\frac{y x}{3}) + 3 (- \frac{y x}{3} cos (\frac{y x}{3}) + sin (\frac{y x}{3})))

Simplify

y^{2} \frac{3}{y ^{2}} (- y cos (\frac{y x}{3}) + 3 (- \frac{y x}{3} cos (\frac{y x}{3}) + sin (\frac{y x}{3}))) : 3 (- y cos (\frac{y x}{3}) - y x cos (\frac{y x}{3}) + 3 sin (\frac{y x}{3}))

= 3 (- y cos (\frac{y x}{3}) - y x cos (\frac{y x}{3}) + 3 sin (\frac{y x}{3}))

Add a constant to the solution

= 3 (- y cos (\frac{y x}{3}) - y x cos (\frac{y x}{3}) + 3 sin (\frac{y x}{3})) + C

n = 1 \sum \infty \frac{n - 5}{n + 5}

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

s l o p e \frac{4 x}{x - 5}

\frac{d x}{d r} + e^{r - x} = 0

\frac{\partial}{\partial x} (e^{8 y} sin (- 8 x))

What is the integral of (x+1)y^2sin((xy)/3) ?
The integral of (x+1)y^2sin((xy)/3) is 3(-ycos((yx)/3)-yxcos((yx)/3)+3sin((yx)/3))+C