Evaluate the integral. int((4)/(x^(2))+(24)/(x^(4)))dx Answer:

Given integral: We are given the integral: int((4)/(x^(2))+(24)/(x^(4)))dx We can split this integral into two separate integrals: int(4/x^2)dx + int(24/x^4)dxSplit into two: Now we will integrate the first part: int(4/x^2)dx This is the same as: int(4x^(-2))dx The integral of x^n is (x^(n+1))/(n+1) for n ≠ -1, so we apply this rule: 4 * int(x^(-2))dx = 4 * (-x^(-1)) = -4/xIntegrate first part: Next, we will integrate the second part: int(24/x^4)dx This is the same as: int(24x^(-4))dx Again, we use the power rule for integration: 24 * int(x^(-4))dx = 24 * (-1/3)x^(-3) = -8x^(-3) = -8/x^3Integrate second part: Now we combine the results of the two integrals and add the constant of integration C: -4/x - 8/x^3 + C

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Evaluate the integral.

int((4)/(x^(2))+(24)/(x^(4)))dx
Answer:

Evaluate the integral. $\newline$ $\int\left(\frac{4}{x^{2}}+\frac{24}{x^{4}}\right) \mathrm{d} x$ $\newline$ Answer:

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Calculus

Find indefinite integrals using the substitution

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Q. Evaluate the integral.

\newline

\int\left(\frac{4}{x^{2}}+\frac{24}{x^{4}}\right) \mathrm{d} x

\newline

Answer:

Given integral: We are given the integral: $\newline$ $\int\left(\frac{4}{x^{2}}+\frac{24}{x^{4}}\right)dx$ $\newline$ We can split this integral into two separate integrals: $\newline$ $\int\frac{4}{x^2}dx + \int\frac{24}{x^4}dx$
Split into two: Now we will integrate the first part: $\newline$ $\int \frac{4}{x^2}\,dx$ $\newline$ This is the same as: $\newline$ $\int 4x^{-2}\,dx$ $\newline$ The integral of $x^n$ is $\frac{x^{n+1}}{n+1}$ for $n \neq -1$ , so we apply this rule: $\newline$ $4 \cdot \int x^{-2}\,dx = 4 \cdot (-x^{-1}) = -\frac{4}{x}$
Integrate first part: Next, we will integrate the second part: $\newline$ $\int \frac{24}{x^4}\,dx$ $\newline$ This is the same as: $\newline$ $\int 24x^{-4}\,dx$ $\newline$ Again, we use the power rule for integration: $\newline$ $24 \cdot \int x^{-4}\,dx = 24 \cdot \left(-\frac{1}{3}\right)x^{-3} = -8x^{-3} = -\frac{8}{x^3}$
Integrate second part: Now we combine the results of the two integrals and add the constant of integration $C$ : $-\frac{4}{x} - \frac{8}{x^3} + C$