en

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (x-1)^2sin^2(x)

Solution

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

Solution

\frac{1}{3} x^{3} sin^{2} (x) + \frac{1}{6} x^{3} cos (2 x) - \frac{1}{4} x^{2} sin (2 x) - x^{2} + \frac{x ^{2}}{2} - \frac{1}{4} x cos (2 x) + \frac{1}{2} x sin (2 x) + \frac{1}{2} x - \frac{1}{8} sin (2 x) + \frac{1}{4} cos (2 x) + C

Solution steps

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

Expand

(x - 1)^{2} sin^{2} (x) : x^{2} sin^{2} (x) - 2 x sin^{2} (x) + sin^{2} (x)

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

Apply the Sum Rule:

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

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

\int x^{2} sin^{2} (x) d x = \frac{1}{3} x^{3} sin^{2} (x) + \frac{1}{6} x^{3} cos (2 x) - \frac{1}{4} x^{2} sin (2 x) - \frac{1}{4} x cos (2 x) + \frac{1}{8} sin (2 x)

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

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

= \frac{1}{3} x^{3} sin^{2} (x) + \frac{1}{6} x^{3} cos (2 x) - \frac{1}{4} x^{2} sin (2 x) - \frac{1}{4} x cos (2 x) + \frac{1}{8} sin (2 x) - (x^{2} - \frac{1}{2} x sin (2 x) - \frac{x ^{2}}{2} - \frac{1}{4} cos (2 x)) + \frac{1}{2} (x - \frac{1}{2} sin (2 x))

Simplify

\frac{1}{3} x^{3} sin^{2} (x) + \frac{1}{6} x^{3} cos (2 x) - \frac{1}{4} x^{2} sin (2 x) - \frac{1}{4} x cos (2 x) + \frac{1}{8} sin (2 x) - (x^{2} - \frac{1}{2} x sin (2 x) - \frac{x ^{2}}{2} - \frac{1}{4} cos (2 x)) + \frac{1}{2} (x - \frac{1}{2} sin (2 x)) : \frac{1}{3} x^{3} sin^{2} (x) + \frac{1}{6} x^{3} cos (2 x) - \frac{1}{4} x^{2} sin (2 x) - x^{2} + \frac{x ^{2}}{2} - \frac{1}{4} x cos (2 x) + \frac{1}{2} x sin (2 x) + \frac{1}{2} x - \frac{1}{8} sin (2 x) + \frac{1}{4} cos (2 x)

= \frac{1}{3} x^{3} sin^{2} (x) + \frac{1}{6} x^{3} cos (2 x) - \frac{1}{4} x^{2} sin (2 x) - x^{2} + \frac{x ^{2}}{2} - \frac{1}{4} x cos (2 x) + \frac{1}{2} x sin (2 x) + \frac{1}{2} x - \frac{1}{8} sin (2 x) + \frac{1}{4} cos (2 x)

Add a constant to the solution

= \frac{1}{3} x^{3} sin^{2} (x) + \frac{1}{6} x^{3} cos (2 x) - \frac{1}{4} x^{2} sin (2 x) - x^{2} + \frac{x ^{2}}{2} - \frac{1}{4} x cos (2 x) + \frac{1}{2} x sin (2 x) + \frac{1}{2} x - \frac{1}{8} sin (2 x) + \frac{1}{4} cos (2 x) + C

Popular Examples

tangent of f(x)= 9/(x^2),\at x=3 tangent of

f (x) = \frac{9}{x ^{2}}, at x = 3

(\partial)/(\partial x)(e^{-7y}sin(7x))

\frac{\partial}{\partial x} (e^{- 7 y} sin (7 x))

derivative of (4x/(5sqrt(2))+3)

\frac{d}{d x} (\frac{4 x}{5 2} + 3)

laplacetransform 25t^2 laplace transform

25 t^{2}

derivative of f(t)=7-1/3 t derivative of

f (t) = 7 - \frac{1}{3} t

Frequently Asked Questions (FAQ)

What is the integral of (x-1)^2sin^2(x) ?
The integral of (x-1)^2sin^2(x) is