int_(1)^(4)(4x^(3)-x^(2)+3x+2)dx

Given Integral: We are given the integral to evaluate: int_(1)^(4)(4x^(3)-x^(2)+3x+2)dx The first step is to find the antiderivative (indefinite integral) of the function 4x^3 - x^2 + 3x + 2 with respect to x. To do this, we will apply the power rule for integration, which states that the integral of x^n dx is (1/(n+1))x^(n+1) + C, where C is the constant of integration.Antiderivative Calculation: The antiderivative of 4x^3 is (4/4)x^(3+1) = x^4. The antiderivative of -x^2 is (-1/3)x^(2+1) = -(1/3)x^3. The antiderivative of 3x is (3/2)x^(1+1) = (3/2)x^2. The antiderivative of 2 is 2x. So, the antiderivative of the function is x^4 - (1/3)x^3 + (3/2)x^2 + 2x + C.Evaluate Antiderivative: Now we need to evaluate the antiderivative from x = 1 to x = 4. This is done by calculating the antiderivative at the upper limit of integration and subtracting the antiderivative at the lower limit of integration. We will first calculate the antiderivative at x = 4: F(4) = 4^4 - (1/3)4^3 + (3/2)4^2 + 2*4Calculate at x=4: Plugging in the values, we get: F(4) = 256 - (1/3)(64) + (3/2)(16) + 8 F(4) = 256 - 21.333... + 24 + 8 F(4) = 256 - 21.333... + 32 F(4) = 266.666...Calculate at x=1: Next, we calculate the antiderivative at x = 1: F(1) = 1^4 - (1/3)1^3 + (3/2)1^2 + 2*1 F(1) = 1 - (1/3) + (3/2) + 2Subtract to Find Integral: Plugging in the values, we get: F(1) = 1 - 0.333... + 1.5 + 2 F(1) = 1 - 0.333... + 3.5 F(1) = 4.166...Final Result: Now we subtract F(1) from F(4) to get the value of the definite integral: int_(1)^(4)(4x^(3)-x^(2)+3x+2)dx = F(4) - F(1) int_(1)^(4)(4x^(3)-x^(2)+3x+2)dx = 266.666... - 4.166...Performing the subtraction, we get: int_(1)^(4)(4x^(3)-x^(2)+3x+2)dx = 262.5 This is the exact, simplified answer to the definite integral.

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\int_{1}^{4}\left(4 x^{3}-x^{2}+3 x+2\right) d x

Given Integral: We are given the integral to evaluate: $\int_{1}^{4}(4x^{3}-x^{2}+3x+2)dx$ The first step is to find the antiderivative (indefinite integral) of the function $4x^3 - x^2 + 3x + 2$ with respect to $x$ . To do this, we will apply the power rule for integration, which states that the integral of $x^n dx$ is $\frac{1}{(n+1)}x^{(n+1)} + C$ , where $C$ is the constant of integration.
Antiderivative Calculation: The antiderivative of $4x^3$ is $(\frac{4}{4})x^{(3+1)} = x^4$ . The antiderivative of $-x^2$ is $(-\frac{1}{3})x^{(2+1)} = -(\frac{1}{3})x^3$ . The antiderivative of $3x$ is $(\frac{3}{2})x^{(1+1)} = (\frac{3}{2})x^2$ . The antiderivative of $2$ is $2x$ . So, the antiderivative of the function is $x^4 - (\frac{1}{3})x^3 + (\frac{3}{2})x^2 + 2x + C$ .
Evaluate Antiderivative: Now we need to evaluate the antiderivative from $x = 1$ to $x = 4$ . This is done by calculating the antiderivative at the upper limit of integration and subtracting the antiderivative at the lower limit of integration. $\newline$ We will first calculate the antiderivative at $x = 4$ : $\newline$ $F(4) = 4^4 - (\frac{1}{3})4^3 + (\frac{3}{2})4^2 + 2\cdot 4$
Calculate at $x=4$ : Plugging in the values, we get: $\newline$ $F(4) = 256 - (\frac{1}{3})(64) + (\frac{3}{2})(16) + 8$ $\newline$ $F(4) = 256 - 21.333... + 24 + 8$ $\newline$ $F(4) = 256 - 21.333... + 32$ $\newline$ $F(4) = 266.666...$
Calculate at $x=1$ : Next, we calculate the antiderivative at $x = 1$ :
$F(1) = 1^4 - (\frac{1}{3})1^3 + (\frac{3}{2})1^2 + 2\cdot 1$
$F(1) = 1 - (\frac{1}{3}) + (\frac{3}{2}) + 2$
Subtract to Find Integral: Plugging in the values, we get: $\newline$ $F(1) = 1 - 0.333... + 1.5 + 2$ $\newline$ $F(1) = 1 - 0.333... + 3.5$ $\newline$ $F(1) = 4.166...$
Final Result: Now we subtract $F(1)$ from $F(4)$ to get the value of the definite integral: $\newline$ $\int_{1}^{4}(4x^{3}-x^{2}+3x+2)\,dx = F(4) - F(1)$ $\newline$ $\int_{1}^{4}(4x^{3}-x^{2}+3x+2)\,dx = 266.666\ldots - 4.166\ldots$
Final Result: Now we subtract $F(1)$ from $F(4)$ to get the value of the definite integral: $\newline$ $\int_{1}^{4}(4x^{3}-x^{2}+3x+2)\,dx = F(4) - F(1)$ $\newline$ $\int_{1}^{4}(4x^{3}-x^{2}+3x+2)\,dx = 266.666\ldots - 4.166\ldots$ Performing the subtraction, we get: $\newline$ $\int_{1}^{4}(4x^{3}-x^{2}+3x+2)\,dx = 262.5$ $\newline$ This is the exact, simplified answer to the definite integral.