Evaluate The Integral:$\[ \int_9^{17} \frac{15600}{t^2 - 29t + 160} \, Dt \\]

=====================================================

Introduction

---

In this article, we will delve into the world of calculus and evaluate a given integral. The integral in question is ${ \int_9^{17} \frac{15600}{t^2 - 29t + 160} , dt }$. This integral appears to be a rational function, and our goal is to find its antiderivative and then evaluate it over the given interval.

Breaking Down the Integral

---

To tackle this problem, we first need to analyze the integral and identify its components. The integral is a rational function, which means it is the ratio of two polynomials. In this case, the numerator is a constant, 15600, while the denominator is a quadratic polynomial, $t^2 - 29t + 160$.

Factoring the Denominator

---

Before we proceed, let's try to factor the denominator. We can use the quadratic formula to find the roots of the quadratic equation $t^2 - 29t + 160 = 0$. The quadratic formula states that the roots of the equation $ax^2 + bx + c = 0$ are given by:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

In our case, $a = 1$, $b = -29$, and $c = 160$. Plugging these values into the quadratic formula, we get:

$$t = \frac{29 \pm \sqrt{(-29)^2 - 4(1)(160)}}{2(1)}$$

Simplifying the expression, we get:

$$t = \frac{29 \pm \sqrt{841 - 640}}{2}$$

$$t = \frac{29 \pm \sqrt{201}}{2}$$

The roots of the quadratic equation are $t = \frac{29 + \sqrt{201}}{2}$ and $t = \frac{29 - \sqrt{201}}{2}$.

Partial Fraction Decomposition

---

Now that we have factored the denominator, we can proceed with partial fraction decomposition. We can write the integral as:

$$\int_9^{17} \frac{15600}{(t - \frac{29 + \sqrt{201}}{2})(t - \frac{29 - \sqrt{201}}{2})} \, dt$$

We can then decompose the rational function into partial fractions:

$$\frac{15600}{(t - \frac{29 + \sqrt{201}}{2})(t - \frac{29 - \sqrt{201}}{2})} = \frac{A}{t - \frac{29 + \sqrt{201}}{2}} + \frac{B}{t - \frac{29 - \sqrt{201}}{2}}$$

where $A$ and $B$ are constants to be determined.

Finding the Constants

---

To find the constants $A$ and $B$, we can multiply both sides of the equation by the common denominator:

$$(t - \frac{29 + \sqrt{201}}{2})(t - \frac{29 - \sqrt{201}}{2}) \left( \frac{15600}{(t - \frac{29 + \sqrt{201}}{2})(t - \frac{29 - \sqrt{201}}{2})} \right) = (t - \frac{29 + \sqrt{201}}{2})(t - \frac{29 - \sqrt{201}}{2}) \left( \frac{A}{t - \frac{29 + \sqrt{201}}{2}} + \frac{B}{t - \frac{29 - \sqrt{201}}{2}} \right)$$

Simplifying the equation, we get:

$$15600 = A(t - \frac{29 - \sqrt{201}}{2}) + B(t - \frac{29 + \sqrt{201}}{2})$$

We can then substitute $t = \frac{29 + \sqrt{201}}{2}$ into the equation to solve for $A$:

$$15600 = A(\frac{29 + \sqrt{201}}{2} - \frac{29 - \sqrt{201}}{2})$$

Simplifying the equation, we get:

$$15600 = A(\sqrt{201})$$

$$A = \frac{15600}{\sqrt{201}}$$

We can then substitute $t = \frac{29 - \sqrt{201}}{2}$ into the equation to solve for $B$:

$$15600 = B(\frac{29 - \sqrt{201}}{2} - \frac{29 + \sqrt{201}}{2})$$

Simplifying the equation, we get:

$$15600 = B(-\sqrt{201})$$

$$B = -\frac{15600}{\sqrt{201}}$$

Evaluating the Integral

---

Now that we have found the constants $A$ and $B$, we can proceed to evaluate the integral:

$$\int_9^{17} \frac{15600}{(t - \frac{29 + \sqrt{201}}{2})(t - \frac{29 - \sqrt{201}}{2})} \, dt = \int_9^{17} \left( \frac{\frac{15600}{\sqrt{201}}}{t - \frac{29 + \sqrt{201}}{2}} + \frac{-\frac{15600}{\sqrt{201}}}{t - \frac{29 - \sqrt{201}}{2}} \right) \, dt$$

We can then evaluate the integral using the fundamental theorem of calculus:

$$\int_9^{17} \left( \frac{\frac{15600}{\sqrt{201}}}{t - \frac{29 + \sqrt{201}}{2}} + \frac{-\frac{15600}{\sqrt{201}}}{t - \frac{29 - \sqrt{201}}{2}} \right) \, dt = \left[ \frac{\frac{15600}{\sqrt{201}}}{\ln|t - \frac{29 + \sqrt{201}}{2}|} - \frac{-\frac{15600}{\sqrt{201}}}{\ln|t - \frac{29 - \sqrt{201}}{2}|} \right]_9^{17}$$

Simplifying the expression, we get:

$$\left[ \frac{\frac{15600}{\sqrt{201}}}{\ln|t - \frac{29 + \sqrt{201}}{2}|} - \frac{-\frac{15600}{\sqrt{201}}}{\ln|t - \frac{29 - \sqrt{201}}{2}|} \right]_9^{17} = \left[ \frac{\frac{15600}{\sqrt{201}}}{\ln|17 - \frac{29 + \sqrt{201}}{2}|} - \frac{-\frac{15600}{\sqrt{201}}}{\ln|17 - \frac{29 - \sqrt{201}}{2}|} \right] - \left[ \frac{\frac{15600}{\sqrt{201}}}{\ln|9 - \frac{29 + \sqrt{201}}{2}|} - \frac{-\frac{15600}{\sqrt{201}}}{\ln|9 - \frac{29 - \sqrt{201}}{2}|} \right]$$

Simplifying the expression further, we get:

$$\left[ \frac{\frac{15600}{\sqrt{201}}}{\ln|17 - \frac{29 + \sqrt{201}}{2}|} - \frac{-\frac{15600}{\sqrt{201}}}{\ln|17 - \frac{29 - \sqrt{201}}{2}|} \right] - \left[ \frac{\frac{15600}{\sqrt{201}}}{\ln|9 - \frac{29 + \sqrt{201}}{2}|} - \frac{-\frac{15600}{\sqrt{201}}}{\ln|9 - \frac{29 - \sqrt{201}}{2}|} \right] = \frac{15600}{\sqrt{201}} \left[ \frac{1}{\ln|17 - \frac{29 + \sqrt{201}}{2}|} - \frac{1}{\ln|9 - \frac{29 + \sqrt{201}}{2}|} \right] - \frac{-15600}{\sqrt{201}} \left[ \frac{1}{\ln|17 - \frac{29 - \sqrt{201}}{2}|} - \frac{1}{\ln|9 - \frac{29 - \sqrt{201}}{2}|} \right]$$

Simplifying the expression even further, we get:

\frac{15600}{\sqrt{201}} \left[ \frac{1}{\ln|17 - \frac{29 + \sqrt{201}}{2}|} - \frac{1}{\ln|9 - \frac{29 + \sqrt{201}}{2}|} \right] - \frac{-15600}{\sqrt{201}} \left[ \frac{1}{\ln|17 - \frac{29 - \sqrt{<br/> # Q&A: Evaluating the Integral ===================================== ## Frequently Asked Questions --------------------------- ### Q: What is the integral in question? A: The integral in question is ${ \int_9^{17} \frac{15600}{t^2 - 29t + 160} \, dt \}$. This integral appears to be a rational function, and our goal is to find its antiderivative and then evaluate it over the given interval. ### Q: How do we approach this problem? A: To tackle this problem, we first need to analyze the integral and identify its components. The integral is a rational function, which means it is the ratio of two polynomials. In this case, the numerator is a constant, 15600, while the denominator is a quadratic polynomial, $t^2 - 29t + 160$. ### Q: What is partial fraction decomposition? A: Partial fraction decomposition is a technique used to break down a rational function into simpler fractions. In this case, we can write the integral as: $\frac{15600}{(t - \frac{29 + \sqrt{201}}{2})(t - \frac{29 - \sqrt{201}}{2})} = \frac{A}{t - \frac{29 + \sqrt{201}}{2}} + \frac{B}{t - \frac{29 - \sqrt{201}}{2}}

where $A$ and $B$ are constants to be determined.

Q: How do we find the constants A and B?

A: To find the constants $A$ and $B$, we can multiply both sides of the equation by the common denominator:

$$(t - \frac{29 + \sqrt{201}}{2})(t - \frac{29 - \sqrt{201}}{2}) \left( \frac{15600}{(t - \frac{29 + \sqrt{201}}{2})(t - \frac{29 - \sqrt{201}}{2})} \right) = (t - \frac{29 + \sqrt{201}}{2})(t - \frac{29 - \sqrt{201}}{2}) \left( \frac{A}{t - \frac{29 + \sqrt{201}}{2}} + \frac{B}{t - \frac{29 - \sqrt{201}}{2}} \right)$$

Simplifying the equation, we get:

$$15600 = A(t - \frac{29 - \sqrt{201}}{2}) + B(t - \frac{29 + \sqrt{201}}{2})$$

We can then substitute $t = \frac{29 + \sqrt{201}}{2}$ into the equation to solve for $A$:

$$15600 = A(\frac{29 + \sqrt{201}}{2} - \frac{29 - \sqrt{201}}{2})$$

Simplifying the equation, we get:

$$15600 = A(\sqrt{201})$$

$$A = \frac{15600}{\sqrt{201}}$$

We can then substitute $t = \frac{29 - \sqrt{201}}{2}$ into the equation to solve for $B$:

$$15600 = B(\frac{29 - \sqrt{201}}{2} - \frac{29 + \sqrt{201}}{2})$$

Simplifying the equation, we get:

$$15600 = B(-\sqrt{201})$$

$$B = -\frac{15600}{\sqrt{201}}$$

Q: How do we evaluate the integral?

A: Now that we have found the constants $A$ and $B$, we can proceed to evaluate the integral:

$$\int_9^{17} \frac{15600}{(t - \frac{29 + \sqrt{201}}{2})(t - \frac{29 - \sqrt{201}}{2})} \, dt = \int_9^{17} \left( \frac{\frac{15600}{\sqrt{201}}}{t - \frac{29 + \sqrt{201}}{2}} + \frac{-\frac{15600}{\sqrt{201}}}{t - \frac{29 - \sqrt{201}}{2}} \right) \, dt$$

We can then evaluate the integral using the fundamental theorem of calculus:

$$\int_9^{17} \left( \frac{\frac{15600}{\sqrt{201}}}{t - \frac{29 + \sqrt{201}}{2}} + \frac{-\frac{15600}{\sqrt{201}}}{t - \frac{29 - \sqrt{201}}{2}} \right) \, dt = \left[ \frac{\frac{15600}{\sqrt{201}}}{\ln|t - \frac{29 + \sqrt{201}}{2}|} - \frac{-\frac{15600}{\sqrt{201}}}{\ln|t - \frac{29 - \sqrt{201}}{2}|} \right]_9^{17}$$

Simplifying the expression, we get:

$$\left[ \frac{\frac{15600}{\sqrt{201}}}{\ln|t - \frac{29 + \sqrt{201}}{2}|} - \frac{-\frac{15600}{\sqrt{201}}}{\ln|t - \frac{29 - \sqrt{201}}{2}|} \right]_9^{17} = \left[ \frac{\frac{15600}{\sqrt{201}}}{\ln|17 - \frac{29 + \sqrt{201}}{2}|} - \frac{-\frac{15600}{\sqrt{201}}}{\ln|17 - \frac{29 - \sqrt{201}}{2}|} \right] - \left[ \frac{\frac{15600}{\sqrt{201}}}{\ln|9 - \frac{29 + \sqrt{201}}{2}|} - \frac{-\frac{15600}{\sqrt{201}}}{\ln|9 - \frac{29 - \sqrt{201}}{2}|} \right]$$

Simplifying the expression further, we get:

$$\left[ \frac{\frac{15600}{\sqrt{201}}}{\ln|17 - \frac{29 + \sqrt{201}}{2}|} - \frac{-\frac{15600}{\sqrt{201}}}{\ln|17 - \frac{29 - \sqrt{201}}{2}|} \right] - \left[ \frac{\frac{15600}{\sqrt{201}}}{\ln|9 - \frac{29 + \sqrt{201}}{2}|} - \frac{-\frac{15600}{\sqrt{201}}}{\ln|9 - \frac{29 - \sqrt{201}}{2}|} \right] = \frac{15600}{\sqrt{201}} \left[ \frac{1}{\ln|17 - \frac{29 + \sqrt{201}}{2}|} - \frac{1}{\ln|9 - \frac{29 + \sqrt{201}}{2}|} \right] - \frac{-15600}{\sqrt{201}} \left[ \frac{1}{\ln|17 - \frac{29 - \sqrt{201}}{2}|} - \frac{1}{\ln|9 - \frac{29 - \sqrt{201}}{2}|} \right]$$

Simplifying the expression even further, we get:

$$\frac{15600}{\sqrt{201}} \left[ \frac{1}{\ln|17 - \frac{29 + \sqrt{201}}{2}|} - \frac{1}{\ln|9 - \frac{29 + \sqrt{201}}{2}|} \right] - \frac{-15600}{\sqrt{201}} \left[ \frac{1}{\ln|17 - \frac{29 - \sqrt{201}}{2}|} - \frac{1}{\ln|9 - \frac{29 - \sqrt{201}}{2}|} \right]$$

Q: What is the final answer?

A: The final answer is:

$$\frac{15600}{\sqrt{201}} \left[ \frac{1}{\ln|17 - \frac{29 + \sqrt{201}}{2}|} - \frac{1}{\ln|9 - \frac{29 + \sqrt{201}}{2}|} \right] - \frac{-15600}{\sqrt{201}} \left[ \frac{1}{\ln|17 - \frac{29 - \sqrt{201}}{2}|} - \frac{1}{\ln|9 - \frac{29 - \sqrt{201}}{2}|} \right]$$

This is the final answer to the integral ${ \int_9^{17} \frac{15600}{t^2 - 29t + 160} , dt }$.