Studdy AI Logo
Math  /  Calculus

Question∫121x5dx\int_{1}^{2} \frac{1}{x^{5}} d x

Studdy Solution

STEP 1

1. The integral ∫121x5 dx\int_{1}^{2} \frac{1}{x^{5}} \, dx is a definite integral.
2. We will use the power rule for integration to solve this integral.

STEP 2

1. Rewrite the integrand in a simpler form.
2. Apply the power rule for integration.
3. Evaluate the antiderivative at the bounds.
4. Calculate the definite integral.

STEP 3

Rewrite the integrand 1x5\frac{1}{x^{5}} in a simpler form using negative exponents:
1x5=x−5\frac{1}{x^{5}} = x^{-5}

STEP 4

Apply the power rule for integration, which states that ∫xn dx=xn+1n+1+C\int x^n \, dx = \frac{x^{n+1}}{n+1} + C, where n≠−1n \neq -1:
∫x−5 dx=x−5+1−5+1+C=x−4−4+C=−14x−4+C\int x^{-5} \, dx = \frac{x^{-5+1}}{-5+1} + C = \frac{x^{-4}}{-4} + C = -\frac{1}{4}x^{-4} + C

STEP 5

Evaluate the antiderivative −14x−4-\frac{1}{4}x^{-4} at the bounds 11 and 22:
[−14x−4]12=(−14(2)−4)−(−14(1)−4)\left[-\frac{1}{4}x^{-4}\right]_{1}^{2} = \left(-\frac{1}{4}(2)^{-4}\right) - \left(-\frac{1}{4}(1)^{-4}\right)

STEP 6

Calculate the values:
−14(2)−4=−14⋅116=−164-\frac{1}{4}(2)^{-4} = -\frac{1}{4} \cdot \frac{1}{16} = -\frac{1}{64}
−14(1)−4=−14⋅1=−14-\frac{1}{4}(1)^{-4} = -\frac{1}{4} \cdot 1 = -\frac{1}{4}

STEP 7

Subtract the evaluated values to find the definite integral:
−164−(−14)=−164+14-\frac{1}{64} - \left(-\frac{1}{4}\right) = -\frac{1}{64} + \frac{1}{4}
Convert 14\frac{1}{4} to a fraction with a denominator of 64:
14=1664\frac{1}{4} = \frac{16}{64}
Now perform the subtraction:
−164+1664=1564-\frac{1}{64} + \frac{16}{64} = \frac{15}{64}
The value of the definite integral is:
1564\boxed{\frac{15}{64}}

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