Question

Elavluate- integration of :-
x × cos2x dx by parts


please solve it as soon as possible
​

Answer 1

$\large\underline{\sf{Solution-}}$

$\rm :\longmapsto\:\displaystyle\int\sf \: x \: cos2x \: dx$

We know,

Formula for integration by parts is

$\boxed{ \sf{ \: \displaystyle\int\sf \: uv \: dx = u\displaystyle\int\sf \: vdx -\displaystyle\int\sf \bigg[\dfrac{d}{dx}u\:\bigg(\displaystyle\int\sf \: vdx \bigg)\bigg]dx}}$

We also know,

$\rm :\longmapsto\:\displaystyle\int\sf \: sinx \: dx = - \: cosx + c$

$\rm :\longmapsto\:\displaystyle\int\sf \: cosx \: dx = \: sinx + c$

$\rm :\longmapsto\:\dfrac{d}{dx}x = 1$

Here,

$\rm :\longmapsto\:u = x$

$\rm :\longmapsto\:v = cos2x$

So,

$\rm :\longmapsto\:\displaystyle\int\sf \: x \: cos2x \: dx$

$\rm  \:  =  \: \:x\displaystyle\int\sf \: cos2xdx -\displaystyle\int\sf \bigg[\dfrac{d}{dx}x\:\bigg(\displaystyle\int\sf \: cos2xdx \bigg)\bigg]dx$

$\rm  \:  =  \: \:x\dfrac{sin2x}{2} - \displaystyle\int\sf \: 1 \times \dfrac{sin2x}{2} \: dx$

$\rm  \:  =  \: \:\dfrac{x \: sin2x}{2} - \displaystyle\int\sf \: \dfrac{sin2x}{2} \: dx$

$\rm  \:  =  \: \:\dfrac{x \: sin2x}{2} \: - \: \dfrac{1}{2} \displaystyle\int\sf \: sin2x \: dx$

$\rm  \:  =  \: \:\dfrac{x \: sin2x}{2} \: + \: \dfrac{1}{2} \: \times \dfrac{cos2x}{2} + c$

$\rm  \:  =  \: \:\dfrac{x \: sin2x}{2} \: + \: \dfrac{cos2x}{4} + c$

Hence,

$\boxed{\rm  \:\displaystyle\int\sf \: x \: cos2x \: dx  =  \: \:\dfrac{x \: sin2x}{2} \: + \: \dfrac{cos2x}{4} + c}$

Basic Concept used :-

Integration by Parts

Integration by Parts is a special technique of integration that is used basically when two functions are multiplied together.

✏️See the rule:

• ∫u v dx = u∫v dx −∫u' (∫v dx) dx

Here,

• u is the function u(x)

• v is the function v(x)

• u' is the derivative of the function u(x)

For integration by parts , the ILATE rule is used to choose u and v.

where,

• I - Inverse trigonometric functions

• L -Logarithmic functions

• A - Arithmetic and Algebraic functions

• T - Trigonometric functions

• E- Exponential functions

The alphabet which comes first is choosen as u and other as v.