QuestionFind the integral $\int \frac{1}{x^{2}-8 x+65} d x$ . Choose the correct answer: (A), (B), (C), or (D).

Studdy Solution

STEP 1

Assumptions1. We are asked to find the integral of the function $\frac{1}{x^{}-8 x+65}$ . . The integral is indefinite, so we will have a constant of integration in our final answer.

STEP 2

The given function can be rewritten as $\frac{1}{(x-4)^{2}+49}$ , which is in the form of $\frac{1}{a^{2}+x^{2}}$ . The integral of this form is $\frac{1}{a}\arctan\left(\frac{x}{a}\right)+C$ .

STEP 3

Rewrite the function in the form of $\frac{1}{a^{2}+x^{2}}$ .
$\int \frac{1}{x^{2}-8 x+65} d x = \int \frac{1}{(x-)^{2}+49} dx$

STEP 4

Now, we can see that the function is in the form $\frac{1}{a^{2}+x^{2}}$ where $a=7$ and $x=x-4$ .

STEP 5

Substitute the values of $a$ and $x$ into the formula $\frac{1}{a}\arctan\left(\frac{x}{a}\right)+C$ .
$\int \frac{1}{(x-4)^{2}+49} dx = \frac{1}{7}\arctan\left(\frac{x-4}{7}\right)+C$ So, the integral of the given function is $\frac{1}{7}\arctan\left(\frac{x-4}{7}\right)+C$ .
This corresponds to answer choice ().

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms