Calculate the integral and write the answer in simplest form. int(-6x^(2)+3x+2x^(3))dx Answer:

Given integral function: We are given the integral of a polynomial function: int(-6x^2 + 3x + 2x^3)dx To solve this, we will integrate each term separately using the power rule for integration, which states that the integral of x^n with respect to x is (x^(n+1))/(n+1) for n ≠ -1.Integrating 2x^3: First, we integrate the term 2x^3: int(2x^3)dx = (2/4)x^(3+1) = (1/2)x^4Integrating -6x^2: Next, we integrate the term -6x^2: int(-6x^2)dx = (-6/3)x^(2+1) = -2x^3Integrating 3x: Finally, we integrate the term 3x: int(3x)dx = (3/2)x^(1+1) = (3/2)x^2Combining results: Now, we combine the results of the three integrals and add the constant of integration C: (1/2)x^4 - 2x^3 + (3/2)x^2 + C

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Find indefinite integrals using the substitution

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Q. Calculate the integral and write the answer in simplest form.

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\int\left(-6 x^{2}+3 x+2 x^{3}\right) d x

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Answer:

Given integral function: We are given the integral of a polynomial function: $\int(-6x^2 + 3x + 2x^3)\,dx$ To solve this, we will integrate each term separately using the power rule for integration, which states that the integral of $x^n$ with respect to $x$ is $\frac{x^{(n+1)}}{n+1}$ for $n \neq -1$ .
Integrating $2x^3$ : First, we integrate the term $2x^3$ : $\int(2x^3)\,dx = \left(\frac{2}{4}\right)x^{(3+1)} = \left(\frac{1}{2}\right)x^4$
Integrating $-6x^2$ : Next, we integrate the term $-6x^2$ : $\int(-6x^2)\,dx = \left(-\frac{6}{3}\right)x^{(2+1)} = -2x^3$
Integrating $3x$ : Finally, we integrate the term $3x$ : $\int(3x)\,dx = \left(\frac{3}{2}\right)x^{(1+1)} = \left(\frac{3}{2}\right)x^2$
Combining results: Now, we combine the results of the three integrals and add the constant of integration $C$ : $(1/2)x^4 - 2x^3 + (3/2)x^2 + C$