Evaluate the integral. int(4x^(3)+6x-8)/(x^(3))dx Answer:

Given Integral Simplification: We are given the integral: int((4x^(3)+6x-8)/(x^(3)))dx First, we simplify the integrand by dividing each term in the numerator by x^3: int(4x^(3)/x^(3) + 6x/x^(3) - 8/x^(3))dx This simplifies to: int(4 + 6/x^2 - 8/x^3)dxIntegrating Each Term: Now we integrate each term separately: int(4)dx + int(6/x^2)dx - int(8/x^3)dx The integral of a constant is the constant times x, and the integral of x^n is x^(n+1)/(n+1) for n ≠ -1. So we have: 4x + 6 * int(x^(-2))dx - 8 * int(x^(-3))dxFinal Integration: We can now integrate the remaining terms: 4x + 6 * (-1/x) - 8 * (-1/2x^2) + C Here, we have used the power rule for integration, which states that the integral of x^n is x^(n+1)/(n+1) for n ≠ -1, and we have added the constant of integration C.Expression Simplification: Simplify the expression: 4x - 6/x + 4/x^2 + C This is the antiderivative of the given function.

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Evaluate the integral.

int(4x^(3)+6x-8)/(x^(3))dx
Answer:

Evaluate the integral. $\newline$ $\int \frac{4 x^{3}+6 x-8}{x^{3}} \mathrm{~d} x$ $\newline$ Answer:

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Calculus

Find indefinite integrals using the substitution

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Q. Evaluate the integral.

\newline

\int \frac{4 x^{3}+6 x-8}{x^{3}} \mathrm{~d} x

\newline

Answer:

Given Integral Simplification: We are given the integral: $\int\left(\frac{4x^{3}+6x-8}{x^{3}}\right)dx$ First, we simplify the integrand by dividing each term in the numerator by $x^3$ : $\int\left(\frac{4x^{3}}{x^{3}} + \frac{6x}{x^{3}} - \frac{8}{x^{3}}\right)dx$ This simplifies to: $\int(4 + \frac{6}{x^2} - \frac{8}{x^3})dx$
Integrating Each Term: Now we integrate each term separately: $\newline$ $\int 4 \, dx + \int \frac{6}{x^2} \, dx - \int \frac{8}{x^3} \, dx$ $\newline$ The integral of a constant is the constant times $x$ , and the integral of $x^n$ is $\frac{x^{n+1}}{n+1}$ for $n \neq -1$ . So we have: $\newline$ $4x + 6 \times \int x^{-2} \, dx - 8 \times \int x^{-3} \, dx$
Final Integration: We can now integrate the remaining terms: $4x + 6 \cdot (-1/x) - 8 \cdot (-1/2x^2) + C$ Here, we have used the power rule for integration, which states that the integral of $x^n$ is $x^{(n+1)}/(n+1)$ for $n \neq -1$ , and we have added the constant of integration $C$ .
Expression Simplification: Simplify the expression: $\newline$ $4x - \frac{6}{x} + \frac{4}{x^2} + C$ $\newline$ This is the antiderivative of the given function.