# Integrate the function $\dfrac{1}{x{{\left( \log x \right)}^{m}}},x>0,m\ne 1$

Last updated date: 24th Mar 2023

•

Total views: 308.1k

•

Views today: 5.85k

Answer

Verified

308.1k+ views

Hint:Substitute $t=\log x$, differentiate them and put the values in the integral. Simplify the integral given by the basic integral formula. Replace $t$ with $\log x$ in the simplified integral.

Complete step-by-step answer:

Let’s consider $I=\int{\dfrac{1}{x{{\left( \log x \right)}^{m}}}dx\ldots \ldots (1)}$

Let’s assume that $\log x=t.$

$\begin{align}

& \therefore x={{\log }^{-1}}t={{e}^{t}} \\

& \Rightarrow x={{e}^{t}} \\

\end{align}$

Differentiate $\log x=t$ with respect to $x.$

$\begin{align}

& \log x=t \\

& \Rightarrow \dfrac{1}{x}dx=dt \\

\end{align}$

Now substitute these values in equation (1).

\[\begin{align}

& I=\int{\dfrac{1}{x{{\left( \log x \right)}^{m}}}dx=\int{\dfrac{dt}{{{t}^{m}}}}} \\

& I=\int{\dfrac{dt}{{{t}^{m}}}}=\int{{{t}^{-m}}dt\ldots \ldots (2)} \\

\end{align}\]

Now integrate equation (2).

We know \[\int{{{x}^{1}}dx=\dfrac{{{x}^{1+1}}}{1+1}=\dfrac{{{x}^{2}}}{2}}+c\] where c is a constant of integration.

Similarly, \[I=\int{{{t}^{-m}}=\left( \dfrac{{{t}^{-m+1}}}{-m+1} \right)+c}\]

\[I=\dfrac{{{t}^{1-m}}}{1-m}+c\]

Now replace $t$ with $\log x$, we get

\[\begin{align}

& I=\dfrac{{{\left( \log x \right)}^{1-m}}}{1-m}+c \\

& I=\dfrac{{{\left( \log x \right)}^{1}}\times {{\left( \log x \right)}^{-m}}}{\left( 1-m \right)}+c \\

& I=\dfrac{1}{1-m}\times \dfrac{\log x}{{{\left( \log x \right)}^{m}}}+c \\

\end{align}\]

Note: The integration can be solved by basic integral formula \[\int{{{x}^{1}}dx=\dfrac{{{x}^{1+1}}}{1+1}=\dfrac{{{x}^{2}}}{2}}+c\]

Similarly \[\Rightarrow \int{1.dx=x+c;\int{a.dx=ax+c}}\] etc.

Complete step-by-step answer:

Let’s consider $I=\int{\dfrac{1}{x{{\left( \log x \right)}^{m}}}dx\ldots \ldots (1)}$

Let’s assume that $\log x=t.$

$\begin{align}

& \therefore x={{\log }^{-1}}t={{e}^{t}} \\

& \Rightarrow x={{e}^{t}} \\

\end{align}$

Differentiate $\log x=t$ with respect to $x.$

$\begin{align}

& \log x=t \\

& \Rightarrow \dfrac{1}{x}dx=dt \\

\end{align}$

Now substitute these values in equation (1).

\[\begin{align}

& I=\int{\dfrac{1}{x{{\left( \log x \right)}^{m}}}dx=\int{\dfrac{dt}{{{t}^{m}}}}} \\

& I=\int{\dfrac{dt}{{{t}^{m}}}}=\int{{{t}^{-m}}dt\ldots \ldots (2)} \\

\end{align}\]

Now integrate equation (2).

We know \[\int{{{x}^{1}}dx=\dfrac{{{x}^{1+1}}}{1+1}=\dfrac{{{x}^{2}}}{2}}+c\] where c is a constant of integration.

Similarly, \[I=\int{{{t}^{-m}}=\left( \dfrac{{{t}^{-m+1}}}{-m+1} \right)+c}\]

\[I=\dfrac{{{t}^{1-m}}}{1-m}+c\]

Now replace $t$ with $\log x$, we get

\[\begin{align}

& I=\dfrac{{{\left( \log x \right)}^{1-m}}}{1-m}+c \\

& I=\dfrac{{{\left( \log x \right)}^{1}}\times {{\left( \log x \right)}^{-m}}}{\left( 1-m \right)}+c \\

& I=\dfrac{1}{1-m}\times \dfrac{\log x}{{{\left( \log x \right)}^{m}}}+c \\

\end{align}\]

Note: The integration can be solved by basic integral formula \[\int{{{x}^{1}}dx=\dfrac{{{x}^{1+1}}}{1+1}=\dfrac{{{x}^{2}}}{2}}+c\]

Similarly \[\Rightarrow \int{1.dx=x+c;\int{a.dx=ax+c}}\] etc.

Recently Updated Pages

If ab and c are unit vectors then left ab2 right+bc2+ca2 class 12 maths JEE_Main

A rod AB of length 4 units moves horizontally when class 11 maths JEE_Main

Evaluate the value of intlimits0pi cos 3xdx A 0 B 1 class 12 maths JEE_Main

Which of the following is correct 1 nleft S cup T right class 10 maths JEE_Main

What is the area of the triangle with vertices Aleft class 11 maths JEE_Main

KCN reacts readily to give a cyanide with A Ethyl alcohol class 12 chemistry JEE_Main

Trending doubts

What was the capital of Kanishka A Mathura B Purushapura class 7 social studies CBSE

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Tropic of Cancer passes through how many states? Name them.

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

Name the Largest and the Smallest Cell in the Human Body ?