Question

integration x^4 log x dx​

Answer 1

Answer :

I hope it will help you.

Answer 2

Given:

We have given an equation of integetation which is $\int\limits {x^4 logx} \, dx$.

To Find:

We have to find the integeration of the given equation and simplify it?

Step-by-step explanation:

• We have given an equation of integeration which is given as

        $\int\limits {x^4 logx} \, dx$

• We will find the integeration of above equation by the by parts formula which is given as

• f(x)= First function, g(x)= Second function

• Then by usiing by parts formula we can say that

      $\bf \int\limits {f(x)g(x)} \, dx =f(x)\int\limits {g(x)} \, dx-\int\limits[\frac{d}{dx} {f(x)}\int\limits {g(x)} \, dx )]\,dx$

• We will choose the first function and second function by the rule of I LATE

       Hence, in the given functions f(x)=log x and g(x)=

       Now put the value in the by parts formula writteen above

      $\bf \int\limits {x^4logx} \, dx =logx\int\limits {x^4} \, dx-\int\limits[\frac{d}{dx} {logx}\int\limits {x^4} \, dx )]\,dx$

• We use the formula of intgeration inthe above equation and we get

      $\bf\int\limits {x^4logx} \, dx =logx\times\frac{x^5}{5} -\int\limits {\frac{1}{x} \frac{x^5}{5} } \, dx$

• Simplify the above equation and again use the integeration basic formula

      $\bf\int\limits {x^4logx} \, dx =\frac{x^5}{5}logx -\int\limits { \frac{x^4}{5} } \, dx\\\bf\int\limits {x^4logx} \, dx =\frac{x^5}{5}logx -\frac{1}{5} \int\limits { x^4 } \, dx\\\\\bf\int\limits {x^4logx} \, dx =\frac{x^5}{5}logx -\frac{x^5}{25} +c$

      Where C is the constant of integeration