\int _2^7\left(3x^2-5x+2\right)\:dx
wrong again.
∫[2,7] (3x^2-5x+2) dx = 232.5
To solve the integral $\int_2^7 (3x^2 - 5x + 2) \, dx$, we can use the properties of integrals. We'll begin by finding the antiderivative of each term separately:
For the term $3x^2$, the antiderivative is $x^3$, since the derivative of $x^3$ with respect to $x$ is $3x^2$.
For the term $-5x$, the antiderivative is $-\frac{5}{2}x^2$, since the derivative of $-\frac{5}{2}x^2$ with respect to $x$ is $-5x$.
For the term $2$, the antiderivative is $2x$, since the derivative of $2x$ with respect to $x$ is $2$.
Now, we can compute the definite integral using the Fundamental Theorem of Calculus:
$$\int_2^7 (3x^2 - 5x + 2) \, dx = \left[x^3\right]_2^7 - \left[\frac{5}{2}x^2\right]_2^7 + \left[2x\right]_2^7$$
Substituting the values into each antiderivative:
$$\left[(7)^3\right] - \left[\frac{5}{2}(7)^2\right] + \left[2(7)\right] - \left[(2)^3\right] + \left[\frac{5}{2}(2)^2\right] - \left[2(2)\right]$$
Simplifying:
$$343 - \frac{5}{2}(49) + 14 - 8 + \frac{5}{2}(4) - 4$$
$$= 343 - \frac{245}{2} + 14 - 8 + 10 - 4$$
$$= 343 - \frac{245}{2} + 24 - 12$$
$$= 343 - 122.5 + 12$$
$$= 243.5$$
Therefore, $\int_2^7 (3x^2 - 5x + 2) \, dx = 243.5$.
Apologies for the error. Let's go through the computation again.
To evaluate the integral $\int_2^7 (3x^2 - 5x + 2) \, dx$, we can use the power rule of integration.
First, we find the antiderivative of each term separately:
For the term $3x^2$, the antiderivative is $\frac{1}{3}x^3$.
For the term $-5x$, the antiderivative is $-\frac{5}{2}x^2$.
For the constant term $2$, the antiderivative is $2x$.
Now, we can evaluate the definite integral using the property that the integral of a sum is the sum of the integrals:
$$\int_2^7 (3x^2 - 5x + 2) \, dx = \left[\frac{1}{3}x^3\right]_2^7 - \left[\frac{5}{2}x^2\right]_2^7 + \left[2x\right]_2^7$$
Evaluating the definite integral at the upper and lower limits:
$$\left[\frac{1}{3}(7^3)\right] - \left[\frac{5}{2}(7^2)\right] + \left[2(7)\right] - \left[\frac{1}{3}(2^3)\right] + \left[\frac{5}{2}(2^2)\right] - \left[2(2)\right]$$
Simplifying the expression:
$$\frac{1}{3}(343) - \frac{5}{2}(49) + 14 - \frac{1}{3}(8) + \frac{5}{2}(4) - 4$$
$$= \frac{343}{3} - \frac{245}{2} + 14 - \frac{8}{3} + 10 - 4$$
$$= \frac{343}{3} - \frac{735}{6} + \frac{42}{3} - \frac{8}{3} + 10 - 4$$
$$= \frac{686}{6} - \frac{735}{6} + \frac{42}{3} - \frac{8}{3} + 6$$
$$= \frac{-49}{6} + \frac{84}{3} + 6$$
$$= \frac{-49}{6} + \frac{252}{6} + \frac{36}{6}$$
$$= \frac{-49 + 252 + 36}{6}$$
$$= \frac{239}{6}$$
Thus, $\int_2^7 (3x^2 - 5x + 2) \, dx = \frac{239}{6} \approx 39.83$.