Integrate. int(20000)/((x+100)^(3))dx

Given integral: We are given the integral to evaluate: ∫20000/(x+100)^3 dx First, we will use a substitution method to simplify the integral. Let u = x + 100, then du = dx.Substitution method: Now, we rewrite the integral in terms of u: ∫20000/u^3 duRewrite in terms of u: The integral of 20000/u^3 is the same as 20000 times the integral of u^(-3). We can use the power rule for integration, which states that ∫u^n du = u^(n+1)/(n+1) for n ≠ -1.Apply power rule: Applying the power rule, we get: 20000 * ∫u^(-3) du = 20000 * (u^(-2)/(-2))Simplify expression: Simplify the expression: = -10000 * u^(-2)Substitute back for u: Now, we substitute back for u to get the integral in terms of x: = -10000 * (x + 100)^(-2)Add constant of integration: Finally, we add the constant of integration C to get the indefinite integral: = -10000/(x + 100)^2 + C

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Integrate. $\newline$ $\int \frac{20000}{(x+100)^{3}} d x$

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Q. Integrate.

\newline

\int \frac{20000}{(x+100)^{3}} d x

Given integral: We are given the integral to evaluate: $\newline$ $\int \frac{20000}{(x+100)^3} \, dx$ $\newline$ First, we will use a substitution method to simplify the integral. $\newline$ Let $u = x + 100$ , then $du = dx$ .
Substitution method: Now, we rewrite the integral in terms of $u$ : $\int \frac{20000}{u^3} \, du$
Rewrite in terms of $u$ : The integral of $\frac{20000}{u^3}$ is the same as $20000$ times the integral of $u^{-3}$ . We can use the power rule for integration, which states that $\int u^n \, du = \frac{u^{n+1}}{n+1}$ for $n \neq -1$ .
Apply power rule: Applying the power rule, we get: $20000 \times \int u^{-3} \, du = 20000 \times \left(\frac{u^{-2}}{-2}\right)$
Simplify expression: Simplify the expression: $\newline$ $= -10000 \cdot u^{-2}$
Substitute back for u: Now, we substitute back for u to get the integral in terms of $x$ : $= -10000 \times (x + 100)^{-2}$
Add constant of integration: Finally, we add the constant of integration $C$ to get the indefinite integral: $\newline$ $= -\frac{10000}{(x + 100)^2} + C$