Question

Evaluate this question​

Answer 1

Answer:

Step-by-step explanation:

65x ^3

Answer 2

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

Given integral is

$\rm :\longmapsto\:\displaystyle\int\sf {x}^{2} \: cosx \: dx$

Let assume that

$\rm :\longmapsto\:I \: = \: \displaystyle\int\sf {x}^{2} \: cosx \: dx$

We know,

Integration by Parts,

$\rm :\longmapsto\:\displaystyle\int\sf \: uvdx = u\displaystyle\int\sf \: vdx - \displaystyle\int\sf \: \bigg(\dfrac{d}{dx}u\displaystyle\int\sf \: vdx\bigg)dx$

Here,

$\rm :\longmapsto\:u = {x}^{2}$

and

$\rm :\longmapsto\:v = cosx$

So,

$\rm :\longmapsto\:I \: = \: \displaystyle\int\sf {x}^{2} \: cosx \: dx$

$\rm \: = \: \: {x}^{2} \displaystyle\int\sf \: cosxdx - \displaystyle\int\sf \: \bigg(\dfrac{d}{dx} {x}^{2} \displaystyle\int\sf \: cosxdx\bigg)dx$

We know,

$\boxed{ \bf{ \: \displaystyle\int\sf \: cosxdx = sinx + c}}$

and

$\boxed{ \sf{ \: \dfrac{d}{dx} {x}^{2} = 2x }}$

So, using these

$\rm \: = \: \: {x}^{2}sinx - \displaystyle\int\sf2x \: sinx \: dx$

$\rm \: = \: \: {x}^{2}sinx - 2\bigg \{ x\displaystyle\int\sf \: sinxdx - \displaystyle\int\sf \: \bigg(\dfrac{d}{dx}x\displaystyle\int\sf \: sinxdx\bigg)dx\bigg \}$

We know,

$\boxed{ \bf{ \: \displaystyle\int\sf \: sinxdx = - \: cosx + c}}$

and

$\boxed{ \sf{ \: \dfrac{d}{dx} {x} = 1 }}$

$\rm \: = \: \: {x}^{2}sinx - 2\bigg \{ - x \: cosx + \displaystyle\int\sf \:cosx \: dx \bigg \}$

$\rm \: = \: \: {x}^{2}sinx - 2\bigg \{ - x \: cosx + sinx \bigg \} + c$

$\rm \: = \: \: {x}^{2}sinx + 2x \: cosx \: - \:2sinx + c$

Hence,

$\boxed{ \rm{\displaystyle\int\bf {x}^{2}cosxdx = \: \: {x}^{2}sinx + 2x \: cosx \: - 2sinx + c}}$

Basic Information :-

Integration by Parts

✏️See the rule:

$\rm :\longmapsto\:\displaystyle\int\sf \: uvdx = u\displaystyle\int\sf \: vdx - \displaystyle\int\sf \: \bigg(\dfrac{d}{dx}u\displaystyle\int\sf \: vdx\bigg)dx$

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.