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

Identify Integral: Identify the integral to be solved. We need to integrate the function -6x^(-3) + 2x with respect to x.Break Down Integral: Break down the integral into two separate integrals. The integral of a sum of functions is the sum of the integrals of each function, so we can write: ∫(-6x^(-3) + 2x)dx = ∫(-6x^(-3))dx + ∫(2x)dxIntegrate First Part: Integrate the first part of the function, -6x^(-3). The integral of x^n with respect to x is (x^(n+1))/(n+1) + C, where n ≠ -1. Applying this rule: ∫(-6x^(-3))dx = -6 * ∫(x^(-3))dx = -6 * (x^(-3+1)/(-3+1)) + C = -6 * (x^(-2)/(-2)) + C = 3x^(-2) + C1Integrate Second Part: Integrate the second part of the function, 2x. Using the same rule as before: ∫(2x)dx = 2 * ∫(x^1)dx = 2 * (x^(1+1)/(1+1)) + C = 2 * (x^2/2) + C = x^2 + C2Combine Results: Combine the results of the two integrals. The combined result of the integrals is: 3x^(-2) + C1 + x^2 + C2 Since C1 and C2 are both constants, we can combine them into a single constant C. Final result: 3x^(-2) + x^2 + C

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Calculate the integral and write the answer in simplest form.

int(-6x^(-3)+2x)dx
Answer:

Calculate the integral and write the answer in simplest form. $\newline$ $\int\left(-6 x^{-3}+2 x\right) d x$ $\newline$ Answer:

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Calculus

Evaluate definite integrals using the power rule

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

\newline

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

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Answer:

Identify Integral: Identify the integral to be solved. $\newline$ We need to integrate the function $-6x^{-3} + 2x$ with respect to $x$ .
Break Down Integral: Break down the integral into two separate integrals. $\newline$ The integral of a sum of functions is the sum of the integrals of each function, so we can write: $\newline$ \int(\(-6x^{ $-3$ } + $2$ x)\,dx = \int( $-6$ x^{ $-3$ })\,dx + \int( $2$ x)\,dx
Integrate First Part: Integrate the first part of the function, $-6x^{-3}$ . The integral of $x^n$ with respect to $x$ is $(x^{n+1})/(n+1) + C$ , where $n \neq -1$ . Applying this rule: $\int(-6x^{-3})dx = -6 \cdot \int(x^{-3})dx = -6 \cdot (x^{-3+1}/(-3+1)) + C = -6 \cdot (x^{-2}/(-2)) + C = 3x^{-2} + C_1$
Integrate Second Part: Integrate the second part of the function, $2x$ . Using the same rule as before: $\int(2x)dx = 2 \times \int(x^1)dx = 2 \times \left(\frac{x^{1+1}}{1+1}\right) + C = 2 \times \left(\frac{x^2}{2}\right) + C = x^2 + C_2$
Combine Results: Combine the results of the two integrals. $\newline$ The combined result of the integrals is: $\newline$ $3x^{-2} + C_1 + x^2 + C_2$ $\newline$ Since $C_1$ and $C_2$ are both constants, we can combine them into a single constant $C$ . $\newline$ Final result: $3x^{-2} + x^2 + C$