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Math  /  Calculus

QuestionConsider the function f(x)=7x4−8x6f(x)=\frac{7}{x^{4}}-\frac{8}{x^{6}}. Let F(x)F(x) be the antiderivative of f(x)f(x) with F(1)=0F(1)=0. Compute F(2)F(2). F(2)=F(2)= □\square Submit answer Next item wers Attempt 2 of 2

Studdy Solution

STEP 1

1. We are given the function f(x)=7x4−8x6 f(x) = \frac{7}{x^4} - \frac{8}{x^6} .
2. F(x) F(x) is the antiderivative of f(x) f(x) .
3. We know that F(1)=0 F(1) = 0 .
4. We need to compute F(2) F(2) .

STEP 2

1. Rewrite f(x) f(x) in a form suitable for integration.
2. Find the antiderivative F(x) F(x) .
3. Use the initial condition F(1)=0 F(1) = 0 to solve for the constant of integration.
4. Evaluate F(2) F(2) .

STEP 3

Rewrite f(x) f(x) in terms of negative exponents:
f(x)=7x−4−8x−6 f(x) = 7x^{-4} - 8x^{-6}

STEP 4

Integrate f(x) f(x) to find F(x) F(x) :
F(x)=∫(7x−4−8x−6) dx F(x) = \int (7x^{-4} - 8x^{-6}) \, dx
Integrate each term separately:
F(x)=7∫x−4 dx−8∫x−6 dx F(x) = 7 \int x^{-4} \, dx - 8 \int x^{-6} \, dx

STEP 5

Compute the integrals:
∫x−4 dx=x−3−3=−13x−3 \int x^{-4} \, dx = \frac{x^{-3}}{-3} = -\frac{1}{3}x^{-3}
∫x−6 dx=x−5−5=−15x−5 \int x^{-6} \, dx = \frac{x^{-5}}{-5} = -\frac{1}{5}x^{-5}
Substitute back into the expression for F(x) F(x) :
F(x)=7(−13x−3)−8(−15x−5) F(x) = 7 \left(-\frac{1}{3}x^{-3}\right) - 8 \left(-\frac{1}{5}x^{-5}\right)
F(x)=−73x−3+85x−5+C F(x) = -\frac{7}{3}x^{-3} + \frac{8}{5}x^{-5} + C

STEP 6

Use the initial condition F(1)=0 F(1) = 0 to solve for C C :
F(1)=−73(1)−3+85(1)−5+C=0 F(1) = -\frac{7}{3}(1)^{-3} + \frac{8}{5}(1)^{-5} + C = 0
−73+85+C=0 -\frac{7}{3} + \frac{8}{5} + C = 0
Find a common denominator and solve for C C :
−3515+2415+C=0 -\frac{35}{15} + \frac{24}{15} + C = 0
−1115+C=0 -\frac{11}{15} + C = 0
C=1115 C = \frac{11}{15}

STEP 7

Substitute C C back into F(x) F(x) :
F(x)=−73x−3+85x−5+1115 F(x) = -\frac{7}{3}x^{-3} + \frac{8}{5}x^{-5} + \frac{11}{15}
Evaluate F(2) F(2) :
F(2)=−73(2)−3+85(2)−5+1115 F(2) = -\frac{7}{3}(2)^{-3} + \frac{8}{5}(2)^{-5} + \frac{11}{15}
Calculate each term:
(2)−3=18,(2)−5=132 (2)^{-3} = \frac{1}{8}, \quad (2)^{-5} = \frac{1}{32}
F(2)=−73⋅18+85⋅132+1115 F(2) = -\frac{7}{3} \cdot \frac{1}{8} + \frac{8}{5} \cdot \frac{1}{32} + \frac{11}{15}
F(2)=−724+120+1115 F(2) = -\frac{7}{24} + \frac{1}{20} + \frac{11}{15}
Find a common denominator and compute:
F(2)=−35120+6120+88120 F(2) = -\frac{35}{120} + \frac{6}{120} + \frac{88}{120}
F(2)=−35+6+88120 F(2) = \frac{-35 + 6 + 88}{120}
F(2)=59120 F(2) = \frac{59}{120}
The value of F(2) F(2) is:
59120 \boxed{\frac{59}{120}}

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