Math  /  Calculus

QuestionDetermine whether the improper integral is convergent or divergent. If the improper integral is convergent, evaluate. 23x2dx\int_{2}^{\infty} \frac{3}{x^{2}} d x

Studdy Solution

STEP 1

1. The integral is improper because it has an infinite limit of integration.
2. We need to determine if the integral converges (has a finite value) or diverges (is infinite or undefined).
3. If the integral converges, we will evaluate it to find its value.

STEP 2

1. Set up the integral with a limit to handle the improper nature.
2. Evaluate the integral.
3. Take the limit as the upper bound approaches infinity.
4. Determine convergence or divergence.
5. If convergent, state the value of the integral.

STEP 3

To handle the improper integral, we replace the upper limit of infinity with a variable, say b b , and then take the limit as b b approaches infinity:
23x2dx=limb2b3x2dx \int_{2}^{\infty} \frac{3}{x^{2}} \, dx = \lim_{b \to \infty} \int_{2}^{b} \frac{3}{x^{2}} \, dx

STEP 4

Evaluate the integral 3x2dx \int \frac{3}{x^{2}} \, dx . First, rewrite the integrand:
3x2=3x2 \frac{3}{x^{2}} = 3x^{-2}
The antiderivative of 3x2 3x^{-2} is:
3x2dx=3x+C \int 3x^{-2} \, dx = -\frac{3}{x} + C

STEP 5

Substitute the antiderivative back into the definite integral and evaluate from 2 to b b :
2b3x2dx=[3x]2b=3b+32 \int_{2}^{b} \frac{3}{x^{2}} \, dx = \left[ -\frac{3}{x} \right]_{2}^{b} = -\frac{3}{b} + \frac{3}{2}
Now, take the limit as b b \to \infty :
limb(3b+32) \lim_{b \to \infty} \left( -\frac{3}{b} + \frac{3}{2} \right)

STEP 6

Evaluate the limit:
As b b \to \infty , 3b0 -\frac{3}{b} \to 0 . Therefore, the limit becomes:
limb(3b+32)=0+32=32 \lim_{b \to \infty} \left( -\frac{3}{b} + \frac{3}{2} \right) = 0 + \frac{3}{2} = \frac{3}{2}

STEP 7

Since the limit exists and is finite, the improper integral is convergent.

STEP 8

The value of the convergent integral is 32 \boxed{\frac{3}{2}} .

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