Math  /  Calculus

QuestionEvaluate the integral. sec(9x5)tan(9x5)dx\int \sec (9 x-5) \tan (9 x-5) d x
Determine a change of variables from x to u . Choose the correct choice below.

Studdy Solution

STEP 1

What is this asking? We need to find the integral of a trigonometric function by using a clever substitution! Watch out! Don't forget the constant of integration, and make sure to substitute back to the original variable, *x*, after integrating with respect to *u*!

STEP 2

1. Define the substitution
2. Compute the derivative of the substitution
3. Rewrite the integral
4. Integrate
5. Substitute back

STEP 3

Let's **define** our substitution!
We'll let u=9x5u = 9x - 5.
This seems like a good choice because 9x59x - 5 is inside both the secant and tangent functions.

STEP 4

Now, we need to find dudx\frac{du}{dx}.
Remember, we're taking the derivative of *u* with respect to *x*.
So, dudx=ddx(9x5)=9\frac{du}{dx} = \frac{d}{dx}(9x - 5) = 9.

STEP 5

Next, we'll solve for dxdx.
We can rewrite dudx=9\frac{du}{dx} = 9 as du=9dxdu = 9 \cdot dx.
Dividing both sides by **9**, we get dx=19dudx = \frac{1}{9} \cdot du.

STEP 6

Now, we can **rewrite** the original integral in terms of *u*: sec(9x5)tan(9x5)dx=sec(u)tan(u)(19du)=19sec(u)tan(u)du\int \sec(9x - 5) \tan(9x - 5) dx = \int \sec(u) \tan(u) \left(\frac{1}{9} du\right) = \frac{1}{9} \int \sec(u) \tan(u) du We can pull the constant 19\frac{1}{9} out of the integral, making it easier to work with.

STEP 7

Remember that the derivative of sec(u)\sec(u) is sec(u)tan(u)\sec(u) \tan(u).
So, the integral of sec(u)tan(u)\sec(u) \tan(u) is simply sec(u)\sec(u).
Don't forget to add the constant of integration, which we'll call *C*. 19sec(u)tan(u)du=19sec(u)+C\frac{1}{9} \int \sec(u) \tan(u) du = \frac{1}{9} \sec(u) + C

STEP 8

Finally, we **substitute** u=9x5u = 9x - 5 back into our result: 19sec(u)+C=19sec(9x5)+C\frac{1}{9} \sec(u) + C = \frac{1}{9} \sec(9x - 5) + C

STEP 9

The integral of sec(9x5)tan(9x5)\sec(9x - 5) \tan(9x - 5) with respect to *x* is 19sec(9x5)+C\frac{1}{9} \sec(9x - 5) + C, where *C* is the constant of integration.
The correct substitution is u=9x5u = 9x - 5.

Was this helpful?

Studdy solves anything!

banner

Start learning now

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

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord