Q) How to Integrate $\int \frac{3x^{4}+5x^{3}+7x^{2}+2x+3}{(x-6)^{5}}dx$ ?
First of all let me tell what I think about this question.
In my Coaching Institute, the chapter 'Integration' is over. This question came in my mind while I was solving the questions of 'Integration By Partial Fraction Decomposition'.
Let me give two examples:
Example 1)
Let's integrate $\int\frac{x-5}{(x-7)^{2}}dx$
Now let me tell the solution of $\int\frac{x-5}{(x-7)^{2}}dx$
Let $I=\int\frac{x-5}{(x-7)^{2}}dx$
$\implies \frac{(x-5)}{(x-7)^{2}}=\frac{A}{(x-7)}+\frac{B}{(x-7)^{2}}$
$\implies (x-5)=Ax+(B-7A)$
Upon solving we get :
$A=1, B=2$
$\implies I=\int\frac{1}{(x-7)}dx+\int\frac{2}{(x-7)^{2}}dx$
Finally, after this step, it is easy to solve.
Now let me give the $2^{nd}$ example:
Evaluate $ I_1=\int\frac{3x^{2}+2x+4}{(x-7)^{3}}dx$
Similarly we can integrate this expression by using Partial Fraction Decomposition.
$\implies \frac{3x^{2}+2x+4}{(x-7)^{3}}=\frac{A}{(x-7)}+\frac{B}{(x-7)^{2}}+\frac{C}{(x-7)^{3}}$
$\implies (3x^{2}+2x+4)=A(x-7)^{2}+B(x-7)+C$
Upon solving we get: $A=3,B=44,C=165$
$\implies I_1=\int\frac{3}{(x-7)}dx+\int\frac{44}{(x-7)^{2}}dx+\int\frac{165}{(x-7)^{3}}dx$
After this step it is easy to integrate $I_1$.
Doubt:
But I can't understand how to integrate by using Integration By Partial Fraction Decomposition for higher powers of $x$. For e.g.:
If we want to integrate
$\int\frac{3x^{5}+8x^{4}+6x^{3}+4x^{2}+5x+4}{(x-6)^{6}}$ by using Partial Fraction Decomposition then it will be a very difficult task. Similarly
Integration of $\int \frac{3x^{4}+5x^{3}+7x^{2}+2x+3}{(x-6)^{5}}dx$ by using Partial Fraction Decomposition will be a very difficult task. It will consume huge amount of time. Is there any alternative method to integrate such types of expressions without using Partial Fraction Decomposition ?
Best Answer
Hint: in this particular case there is a shortcut. Substitute $y=x-6$. Then
$\int \frac{3x^{4}+5x^{3}+7x^{2}+2x+3}{(x-6)^{5}}dx= \\ \int \frac{3(y^4+24y^3+216y^2+864y+1296)+5(y^3+18y^2+108y+216)+7(y^2+12y+36)+2(y+6)+3x^{4}+3}{y^{5}}dy$
Divide each term by $y^5$ for easy integration.