Question

1. State the evaluation theorem of calculus also evaluate ∫ log x dx.

Answer 1

Step-by-step explanation:

We have,

$\displaystyle \tt{\int \: log(x) \: dx }$

$\displaystyle \sf{ = log(x)\int dx - \int \bigg \{ \dfrac{d}{dx}(log(x)) \int \: dx \bigg \}\: dx}$

$\displaystyle \sf{ = log(x) \cdot \: x - \int \bigg \{ \dfrac{1}{x}\cdot \: x \bigg \}\: dx}$

$\displaystyle \sf{ = log(x) \cdot \: x - \int dx}$

$\displaystyle \sf{ = log(x) \cdot \: x - x +C}$

$\displaystyle \sf{ = x \{log(x) - 1 \} +C}$

Answer 2

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

Given expression is

$\rm :\longmapsto\:\displaystyle\int\rm logx \: dx$

can be rewritten as

$\rm \:  =  \:\displaystyle\int\rm logx \: . \: 1 \: dx$

We know,

Integration by Parts

$\red{ \boxed{\displaystyle\int\rm uvdx = u\displaystyle\int\rm vdx -\displaystyle\int\rm \bigg[\dfrac{d}{dx}u\displaystyle\int\rm vdx\bigg]dx}}$

So, here,

$\red{\rm :\longmapsto\:u \: = \: logx \: }$

$\red{\rm :\longmapsto\:v \: = \: 1 \: }$

So, on substituting the values in above formula, we get

$\rm \: = \: logx\displaystyle\int\rm 1dx - \displaystyle\int\rm \bigg[\dfrac{d}{dx}logx\displaystyle\int\rm 1 \: dx\bigg]dx$

We know

$\boxed{ \tt{ \: \dfrac{d}{dx}x = 1}} \: \: and \: \: \boxed{ \tt{ \: \displaystyle\int\rm {x}^{n}dx = \frac{ {x}^{n + 1} }{n + 1} + c}}$

So, on using this, we get

$\rm \:  =  \:logx \times x - \displaystyle\int\rm \frac{1}{x} \times x \: dx$

$\rm \:  =  \:xlogx - \displaystyle\int\rm 1 \: dx$

$\rm \:  =  \:xlogx \: - \: x \: + \: c$

$\rm \:  =  \:x(logx \: - \: 1)\: + \: c$

Hence,

$\rm :\longmapsto\:\boxed{ \tt{ \: \: \displaystyle\int\rm logx \: dx = x(logx - 1) + c \: \: }}$

Basic Concept Used

Integration by Parts

See the rule:

$\red{ \boxed{\displaystyle\int\rm uvdx = u\displaystyle\int\rm vdx -\displaystyle\int\rm \bigg[\dfrac{d}{dx}u\displaystyle\int\rm vdx\bigg]dx}}$

↝ 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.

More to know :-

$\blue{\begin{gathered}\begin{gathered}\boxed{\begin{array}{c|c} \bf f(x) & \bf \displaystyle \int \rm \:f(x) \: dx\\ \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf k & \sf kx + c \\ \\ \sf sinx & \sf - \: cosx+ c \\ \\ \sf cosx & \sf \: sinx + c\\ \\ \sf {sec}^{2} x & \sf tanx + c\\ \\ \sf {cosec}^{2}x & \sf - cotx+ c \\ \\ \sf secx \: tanx & \sf secx + c\\ \\ \sf cosecx \: cotx& \sf - \: cosecx + c\\ \\ \sf tanx & \sf logsecx + c\\ \\ \sf \dfrac{1}{x} & \sf logx+ c\\ \\ \sf {e}^{x} & \sf {e}^{x} + c\end{array}} \\ \end{gathered}\end{gathered}}$