Integral of X Sinx

The term integral may also refer to the notion of anti-derivative, a function F whose derivative is the given function ƒ. In this case it is called an indefinite integral, while the integrals discussed in this article are termed definite integrals. Integrate the function f(x) can be written as`int ` f(x) dx. The integral is used to find the surface area, volume geometric solids by using stokes theorem, divergence theorem (like double integral, triple integral).
Source Wikipedia.

Integral Formulas:

1. `int` x n dx = `(x^n+1) / (n+1) ` + c

2. `int`sin x. dx = - cos x + c

3.` int ` cos x .dx = sin x. + c

4.  ` int` sec2x. dx = tan x + c

5. `int` cosec x.cot x. dx = -cosec x + c

6.`int dx / sqrt(a^2 - x^2) = sin^-1(x / a) + c`

Integral Problems:

Integral problem 1:

Find the integration of  x sin x with respect to x.

Solution:

Given term is x sin x

Integral of x sin x can be expressed as `int` x sin x dx

let u = x                         dv = sin x dx

`(du)/(dx)` = 1            and        v = - cos x

We know `int ` u dv  = uv - `int` v du

= x . (-cos x ) - ` int` (-cos x) 1.dx

= - x cos x +` int` cos x dx

= - x cos x + sin x + c

Answer:  `int` x sin x dx  = - x cos x + sin x + c

Integral problem 2:

Find the integration of  (2x + sin x) with respect to x.

Solution:

Given term is 2x + sin x

Integral of (2x + sin x )can be expressed as `int`(2x + sin x )  dx

`int`(2x + sin x )  dx =  `int` 2x dx +` int` (sin x).dx

= 2 `int` x dx +` int`sin x dx

= 2` (x^2/2)`+ (-cos x) + c

= x2 - cos x + c

Answer:  `int` (2x + sin x) dx  =   x2 - cos x + c

Integral problem 3:

Find the integration of  (sin x  + 3cosec2x)  with respect to x

Solution:

Given term is  (sin x  + 3cosec2x)

Integral of (sin x  + 3cosec2x) can be expressed as  `int ` (sin x + 3cosec2x) dx

`int ` (sin x + 3cosec2x) dx  =` int` (sin x dx) +` int` (3cosec2x) dx

= `int` (sin x dx) + 3` int ` (cosec2x) dx

=  (- cos x) + 3 (-cot x) + c

= - cos x - 3 cot x + c

Answer: `int `(sin x  + 3cosec2x) dx  = - (cos x + 3 cot x) + c

Integral problem 4:

Find the integration of  x2 sin x with respect to x.

Solution:

Given term is x2 sin x

Integral of x2 sin x can be expressed as `int`x2 sin x  dx

let u = x2                          dv = sin x dx

`(du)/(dx)` = 2x            and        v = - cos x

We know `int ` u dv  = uv - `int` v du

`int`x2 sin x  dx   = x2 . (-cos x ) - ` int` (-cos x) 2x.dx

= - x2 cos x +` int` 2x cos x dx

=  - x2 cos x + 2` int` x cos x dx + c

= - x2 cos x + 2 I1+c

Take I1 = ` int` x cos x dx

let u = x                       and     dv = cos x dx

`(du)/(dx)` = 1                                   v =  sin x

We know `int ` u dv  = uv - `int` v du

I1 =` int` x cos x dx = x sin x - `int` sin x dx

= x sin x - (-cos x) + c

I1 = x sin x + cos x + c

`int`x2 sin x  dx  =  - x2 cos x + 2 I1+ c

=  - x2 cos x + 2 (x sin x + cos x )+ c

= - x2 cos x + 2x sin x + 2cos x + c

= 2x sin x - (x2 - 2 )cos x + c

Answer:  `int`x2 sin x  dx   =  2x sin x - (x2 - 2 )cos x + c