Math  /  Calculus

QuestionEvaluate exactly, using the Fundamental Theorem of Calculus: 0c(x67+2x)dx=\int_{0}^{c}\left(\frac{x^{6}}{7}+2 x\right) d x= \square

Studdy Solution
Evaluate the definite integral by substituting the limits into the antiderivative:
F(c)=c749+c2 F(c) = \frac{c^7}{49} + c^2 F(0)=0749+02=0 F(0) = \frac{0^7}{49} + 0^2 = 0
Thus, the integral evaluates to:
0c(x67+2x)dx=(c749+c2)0 \int_{0}^{c} \left(\frac{x^6}{7} + 2x\right) \, dx = \left(\frac{c^7}{49} + c^2\right) - 0
0c(x67+2x)dx=c749+c2 \int_{0}^{c} \left(\frac{x^6}{7} + 2x\right) \, dx = \frac{c^7}{49} + c^2
The exact value of the integral is:
c749+c2 \boxed{\frac{c^7}{49} + c^2}

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