Studdy AI Logo
Math  /  Calculus

QuestionUse a change of variables or the table of general integration formulas to evaluate the following definite integral. ∫1412dxx16x2−1\int_{\frac{1}{4}}^{\frac{1}{2}} \frac{d x}{x \sqrt{16 x^{2}-1}}
Click to view the table of general integration formulas. 12\frac{1}{2} ∫14dxx16x2−1=\int_{\frac{1}{4}} \frac{d x}{x \sqrt{16 x^{2}-1}}= □\square (Type an exact answer.)

Studdy Solution

STEP 1

What is this asking? We need to compute a definite integral, which means we're finding the area under a curve between two points, 14\frac{1}{4} and 12\frac{1}{2}.
The function we're integrating looks a little scary, but we've got this! Watch out! Don't forget to plug in the **upper and lower limits** of integration after finding the antiderivative.
Also, be careful with those square roots and make sure to simplify your answer completely!

STEP 2

1. Simplify the integrand
2. Use the table of integrals
3. Evaluate the definite integral

STEP 3

Let's rewrite the expression inside the square root to make it easier to work with.
We've got 16x2−1\sqrt{16x^2 - 1}, which looks a lot like something involving u2−1\sqrt{u^2 - 1}, which is a form we can find in an integral table.
If we factor out the **16**, we get 16(x2−116)\sqrt{16(x^2 - \frac{1}{16})}, which simplifies to 16⋅x2−116\sqrt{16} \cdot \sqrt{x^2 - \frac{1}{16}}, and finally 4x2−(14)24\sqrt{x^2 - (\frac{1}{4})^2}.

STEP 4

Now, our integral looks like this: ∫1412dxx⋅4x2−(14)2 \int_{\frac{1}{4}}^{\frac{1}{2}} \frac{dx}{x \cdot 4\sqrt{x^2 - (\frac{1}{4})^2}} We can pull the **constant factor** of 14\frac{1}{4} out of the integral, which gives us: 14∫1412dxxx2−(14)2 \frac{1}{4} \int_{\frac{1}{4}}^{\frac{1}{2}} \frac{dx}{x\sqrt{x^2 - (\frac{1}{4})^2}} This is much better!

STEP 5

Looking at our integral, we can see it matches the formula: ∫duuu2−a2=1asec⁡−1∣ua∣+C \int \frac{du}{u\sqrt{u^2 - a^2}} = \frac{1}{a} \sec^{-1} \left| \frac{u}{a} \right| + C In our case, u=xu = x and a=14a = \frac{1}{4}.

STEP 6

Substituting a=14a = \frac{1}{4} into the formula, we get: 14∫1412dxxx2−(14)2=14[114sec⁡−1∣x14∣]1412 \frac{1}{4} \int_{\frac{1}{4}}^{\frac{1}{2}} \frac{dx}{x\sqrt{x^2 - (\frac{1}{4})^2}} = \frac{1}{4} \left[ \frac{1}{\frac{1}{4}} \sec^{-1} \left| \frac{x}{\frac{1}{4}} \right| \right]_{\frac{1}{4}}^{\frac{1}{2}} Simplifying the fraction 114\frac{1}{\frac{1}{4}} by multiplying by 44\frac{4}{4} gives us 41\frac{4}{1}, which is just **4**.

STEP 7

Our expression now looks like this: 14[4sec⁡−1∣4x∣]1412 \frac{1}{4} \left[ 4 \sec^{-1} |4x| \right]_{\frac{1}{4}}^{\frac{1}{2}} Multiplying the 14\frac{1}{4} and **4** gives us **1**, leaving us with: [sec⁡−1∣4x∣]1412 \left[ \sec^{-1} |4x| \right]_{\frac{1}{4}}^{\frac{1}{2}}

STEP 8

Now, we substitute the **upper and lower limits** of integration: sec⁡−1∣4⋅12∣−sec⁡−1∣4⋅14∣ \sec^{-1} |4 \cdot \frac{1}{2}| - \sec^{-1} |4 \cdot \frac{1}{4}| sec⁡−1∣2∣−sec⁡−1∣1∣ \sec^{-1} |2| - \sec^{-1} |1| sec⁡−1(2)−sec⁡−1(1) \sec^{-1}(2) - \sec^{-1}(1)

STEP 9

Since cos⁡(π3)=12\cos(\frac{\pi}{3}) = \frac{1}{2}, we have sec⁡(π3)=2\sec(\frac{\pi}{3}) = 2, so sec⁡−1(2)=π3\sec^{-1}(2) = \frac{\pi}{3}.
Also, sec⁡−1(1)=0\sec^{-1}(1) = 0, because sec⁡(0)=1cos⁡(0)=11=1\sec(0) = \frac{1}{\cos(0)} = \frac{1}{1} = 1. Therefore, our **final answer** is: π3−0=π3 \frac{\pi}{3} - 0 = \frac{\pi}{3}

STEP 10

The value of the definite integral is π3\frac{\pi}{3}.

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