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Step-by-step Solution

Integrate sec(1+x*-4)^3

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Answer

$-\frac{1}{4}\left(\frac{\sec\left(1-4x\right)^{2}\sin\left(1-4x\right)}{2}+\frac{1}{2}\ln\left|\sec\left(1-4x\right)+\tan\left(1-4x\right)\right|\right)+C_0$

Step-by-step explanation

Problem to solve:

$\int\sec^3\left(1-4x\right)dx$
1

Solve the integral $\int\sec\left(1-4x\right)^3dx$ applying u-substitution. Let $u$ and $du$ be

$\begin{matrix}u=1-4x \\ du=-4dx\end{matrix}$
2

Isolate $dx$ in the previous equation

$\frac{du}{-4}=dx$

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Answer

$-\frac{1}{4}\left(\frac{\sec\left(1-4x\right)^{2}\sin\left(1-4x\right)}{2}+\frac{1}{2}\ln\left|\sec\left(1-4x\right)+\tan\left(1-4x\right)\right|\right)+C_0$
$\int\sec^3\left(1-4x\right)dx$

Main topic:

Trigonometric integrals

Used formulas:

6. See formulas

Time to solve it:

~ 0.52 seconds