int3t^(-4)(2+4t^(-3))^(-3)

Identify inner function: Given the integral to solve: I = ∫3t^(-4)(2+4t^(-3))^(-3) dt Identify the inner function for substitution. Let u = 2 + 4t^(-3)Differentiate u: Differentiate u with respect to t to find du/dt. du/dt = d/dt (2 + 4t^(-3)) = 0 - 12t^(-4) = -12t^(-4)Solve for dt: Solve for dt in terms of du and t. dt = du / (-12t^(-4))Substitute u and dt: Substitute u and dt into the integral. I = ∫3t^(-4)(u)^(-3) * (du / (-12t^(-4))) = ∫-1/4 * u^(-3) duIntegrate with respect: Integrate with respect to u. I = -1/4 * ∫u^(-3) du = -1/4 * (-1/2) * u^(-2) = 1/8 * u^(-2)Substitute back for u: Substitute back for u to get the integral in terms of t. I = 1/8 * (2 + 4t^(-3))^(-2)Add constant of integration: Add the constant of integration C to the result. I = 1/8 * (2 + 4t^(-3))^(-2) + C

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Gaussian integral

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\int 3 t^{-4}\left(2+4 t^{-3}\right)^{-3}

Identify inner function: Given the integral to solve: $\newline$ $I = \int 3t^{-4}(2+4t^{-3})^{-3} \, dt$ $\newline$ Identify the inner function for substitution. $\newline$ Let $u = 2 + 4t^{-3}$
Differentiate $u$ : Differentiate $u$ with respect to $t$ to find $\frac{du}{dt}$ . $\frac{du}{dt} = \frac{d}{dt} (2 + 4t^{-3}) = 0 - 12t^{-4} = -12t^{-4}$
Solve for $dt$ : Solve for $dt$ in terms of $du$ and $t$ . $dt = \frac{du}{(-12t^{-4})}$
Substitute $u$ and $dt$ : Substitute $u$ and $dt$ into the integral. $\newline$ $I = \int 3t^{-4}(u)^{-3} * (\frac{du}{-12t^{-4}})$ $\newline$ $= \int -\frac{1}{4} * u^{-3} du$
Integrate with respect: Integrate with respect to $u$ .
$I = -\frac{1}{4} \int u^{-3} \, du$
$= -\frac{1}{4} \cdot (-\frac{1}{2}) \cdot u^{-2}$
$= \frac{1}{8} \cdot u^{-2}$
Substitute back for u: Substitute back for u to get the integral in terms of $t$ . $I = \frac{1}{8} \times \left(2 + 4t^{-3}\right)^{-2}$
Add constant of integration: Add the constant of integration $C$ to the result. $I = \frac{1}{8} \cdot (2 + 4t^{-3})^{-2} + C$