Math / Calculus

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

Studdy Solution

STEP 1

1. The function to be integrated is $f(x) = \frac{x^6}{7} + 2x$ .
2. We will use the Fundamental Theorem of Calculus to evaluate the definite integral.
3. The limits of integration are from 0 to $c$ .

STEP 2

1. Find the antiderivative of the integrand.
2. Apply the Fundamental Theorem of Calculus.
3. Evaluate the definite integral.

STEP 3

First, find the antiderivative of the function $f(x) = \frac{x^6}{7} + 2x$ .
The antiderivative of $\frac{x^6}{7}$ is:
$\int \frac{x^6}{7} \, dx = \frac{1}{7} \cdot \frac{x^{7}}{7} = \frac{x^7}{49}$
The antiderivative of $2x$ is:
$\int 2x \, dx = 2 \cdot \frac{x^2}{2} = x^2$
Thus, the antiderivative of $f(x)$ is:
$F(x) = \frac{x^7}{49} + x^2$

STEP 4

Apply the Fundamental Theorem of Calculus, which states that if $F(x)$ is an antiderivative of $f(x)$ , then:
$\int_{a}^{b} f(x) \, dx = F(b) - F(a)$
In this case, we have:
$\int_{0}^{c} \left(\frac{x^6}{7} + 2x\right) \, dx = F(c) - F(0)$

STEP 5

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

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