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Math  /  Calculus

QuestionFind the family of antiderivatives of the function f(x)=8.5x7F(x)=\begin{array}{l} f(x)=\frac{8.5}{\sqrt{x^{7}}} \\ F(x)= \end{array}

Studdy Solution

STEP 1

1. We are given the function f(x)=8.5x7 f(x) = \frac{8.5}{\sqrt{x^{7}}} .
2. We need to find the family of antiderivatives F(x) F(x) .

STEP 2

1. Simplify the expression for f(x) f(x) .
2. Rewrite the expression in a form suitable for integration.
3. Integrate the function to find the antiderivative.
4. Add the constant of integration.

STEP 3

Simplify the expression for f(x) f(x) :
f(x)=8.5x7=8.5x7/2 f(x) = \frac{8.5}{\sqrt{x^7}} = \frac{8.5}{x^{7/2}}

STEP 4

Rewrite the expression in a form suitable for integration:
f(x)=8.5⋅x−7/2 f(x) = 8.5 \cdot x^{-7/2}

STEP 5

Integrate the function to find the antiderivative:
F(x)=∫8.5⋅x−7/2 dx F(x) = \int 8.5 \cdot x^{-7/2} \, dx

STEP 6

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≠−1 n \neq -1 :
F(x)=8.5⋅x−7/2+1−7/2+1+C F(x) = 8.5 \cdot \frac{x^{-7/2 + 1}}{-7/2 + 1} + C
F(x)=8.5⋅x−5/2−5/2+C F(x) = 8.5 \cdot \frac{x^{-5/2}}{-5/2} + C

STEP 7

Simplify the expression:
F(x)=8.5⋅(−25)⋅x−5/2+C F(x) = 8.5 \cdot \left(-\frac{2}{5}\right) \cdot x^{-5/2} + C
F(x)=−175⋅x−5/2+C F(x) = -\frac{17}{5} \cdot x^{-5/2} + C

STEP 8

Add the constant of integration C C :
The family of antiderivatives is:
F(x)=−175⋅x−5/2+C F(x) = -\frac{17}{5} \cdot x^{-5/2} + C
The family of antiderivatives is:
F(x)=−175⋅x−5/2+C \boxed{F(x) = -\frac{17}{5} \cdot x^{-5/2} + C}

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