Question

Express each improper integral in Exercises15–20 as a sum of limits of proper definite integrals. Do not calculate any integrals or limits; just write them down.

${\int }_1^5\frac{1}{2x^2-10x+12}dx$

Short answer

The integral is ${\int }_1^2\frac{1}{2x^2-10x+12}dx+{\int }_2^{25}\frac{1}{2x^2-10x+12}dx+{\int }_{25}^3\frac{1}{2x^2-10x+12}dx+{\int }_3^5\frac{1}{2x^2-10x+12}dx$.

Step-by-step solution

Step 1. Given information.   

We are given an integrals,

${\int }_1^5\frac{1}{2x^2-10x+12}dx$

Step 2. Graph of function 

Let $f\left(x\right)=\frac{1}{2x^2-10x+12}$,

The graph is as follows,

Step 3. Expressing the Integral. 

It can be seen from the graph that the function has vertical asymptotes at $x=2$and

$x=3$.

Interval of integration contains points $2,3$, which are not in the domain.

That is split the integral at some points, $x=2,x=2.5,x=3$, in order to separately consider the limit as $x\to \infty$.

So,

${\int }_1^5\frac{1}{2x^2-10x+12}dx={\int }_1^2\frac{1}{2x^2-10x+12}dx+{\int }_2^{25}\frac{1}{2x^2-10x+12}dx+{\int }_{25}^3\frac{1}{2x^2-10x+12}dx+{\int }_3^5\frac{1}{2x^2-10x+12}dx$