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int3t^(-4)(2+4t^(-3))^(7)dt=

$\int 3 t^{-4}\left(2+4 t^{-3}\right)^{7} d t=$

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Evaluate definite integrals using the power rule

Full solution

Q.

\int 3 t^{-4}\left(2+4 t^{-3}\right)^{7} d t=

Substitution: Let's do a substitution. Let $u = 2 + 4t^{-3}$ . Then, $du = -12t^{-4} dt$ .
Solve for $dt$ : We need to solve for $dt$ . $dt = -\frac{du}{12t^{-4}}$ .
Substitute and Simplify: Substitute $u$ and $dt$ into the integral. $\int 3t^{-4}(u)^{7} \times \left(-\frac{1}{12t^{-4}}\right) du$ .
Find Antiderivative: Simplify the integral. $\int -\frac{u^{7}}{4} \, du$ .
Simplify Antiderivative: Find the antiderivative of $-\frac{u^{7}}{4}$ . The antiderivative is $-\frac{u^{8}}{(4*8)} + C$ .
Substitute back for u: Simplify the antiderivative. $-u^{8}/32 + C$ .
Simplify Expression: Substitute back for $u$ . $-\left(\frac{2+4t^{-3}}{32}\right)^{8} + C$ .
Simplify Expression: Substitute back for $u$ . $-\left(\frac{(2+4t^{-3})^{8}}{32}\right) + C$ . Simplify the expression. $-\left(\frac{1}{32}\right)(2+4t^{-3})^{8} + C$ .

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