Exercise$\int\sec^3\left(x\right)dx$
Step-by-step Solution
1
Rewrite $\sec\left(x\right)^3$ as the product of two secants
$\int\sec\left(x\right)^2\sec\left(x\right)dx$
2
We can solve the integral $\int\sec\left(x\right)^2\sec\left(x\right)dx$ by applying integration by parts method to calculate the integral of the product of two functions, using the following formula
$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$
Intermediate steps
3
First, identify or choose $u$ and calculate it's derivative, $du$
$\begin{matrix}\displaystyle{u=\sec\left(x\right)}\\ \displaystyle{du=\sec\left(x\right)\tan\left(x\right)dx}\end{matrix}$
Explain this step further
4
Now, identify $dv$ and calculate $v$
$\begin{matrix}\displaystyle{dv=\sec\left(x\right)^2dx}\\ \displaystyle{\int dv=\int \sec\left(x\right)^2dx}\end{matrix}$
5
Solve the integral to find $v$
$v=\int\sec\left(x\right)^2dx$
6
The integral of $\sec(x)^2$ is $\tan(x)$
$\tan\left(x\right)$
Intermediate steps
7
Now replace the values of $u$, $du$ and $v$ in the last formula
$\tan\left(x\right)\sec\left(x\right)-\int\sec\left(x\right)\tan\left(x\right)^2dx$
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Intermediate steps
8
We identify that the integral has the form $\int\tan^m(x)\sec^n(x)dx$. If $n$ is odd and $m$ is even, then we need to express everything in terms of secant, expand and integrate each function separately
$\tan\left(x\right)\sec\left(x\right)-\int\sec\left(x\right)\left(\sec\left(x\right)^2-1\right)dx$
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Intermediate steps
9
Multiply the single term $\sec\left(x\right)$ by each term of the polynomial $\left(\sec\left(x\right)^2-1\right)$
$\tan\left(x\right)\sec\left(x\right)-\int\left(\sec\left(x\right)^{3}-\sec\left(x\right)\right)dx$
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Intermediate steps
10
Simplify the expression
$\tan\left(x\right)\sec\left(x\right)-\int\sec\left(x\right)^{3}dx+\int\sec\left(x\right)dx$
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Intermediate steps
11
The integral $\int\sec\left(x\right)dx$ results in: $\ln\left(\sec\left(x\right)+\tan\left(x\right)\right)$
$\ln\left(\sec\left(x\right)+\tan\left(x\right)\right)$
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12
This integral by parts turned out to be a cyclic one (the integral that we are calculating appeared again in the right side of the equation). We can pass it to the left side of the equation with opposite sign
$\int\sec\left(x\right)^{3}dx=\tan\left(x\right)\sec\left(x\right)-\int\sec\left(x\right)^{3}dx+\ln\left(\sec\left(x\right)+\tan\left(x\right)\right)$
13
Moving the cyclic integral to the left side of the equation
$\int\sec\left(x\right)^{3}dx+\int\sec\left(x\right)^{3}dx=\tan\left(x\right)\sec\left(x\right)+\ln\left|\sec\left(x\right)+\tan\left(x\right)\right|$
$2\int\sec\left(x\right)^{3}dx=\tan\left(x\right)\sec\left(x\right)+\ln\left|\sec\left(x\right)+\tan\left(x\right)\right|$
15
Move the constant term $2$ dividing to the other side of the equation
$\int\sec\left(x\right)^{3}dx=\frac{1}{2}\left(\tan\left(x\right)\sec\left(x\right)+\ln\left|\sec\left(x\right)+\tan\left(x\right)\right|\right)$
16
The integral results in
$\frac{1}{2}\left(\tan\left(x\right)\sec\left(x\right)+\ln\left|\sec\left(x\right)+\tan\left(x\right)\right|\right)$
17
Gather the results of all integrals
$\frac{1}{2}\left(\tan\left(x\right)\sec\left(x\right)+\ln\left|\sec\left(x\right)+\tan\left(x\right)\right|\right)$
18
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
$\frac{1}{2}\left(\tan\left(x\right)\sec\left(x\right)+\ln\left|\sec\left(x\right)+\tan\left(x\right)\right|\right)+C_0$
Intermediate steps
$\frac{1}{2}\tan\left(x\right)\sec\left(x\right)+\frac{1}{2}\ln\left|\sec\left(x\right)+\tan\left(x\right)\right|+C_0$
Explain this step further
Final answer to the exercise $\frac{1}{2}\tan\left(x\right)\sec\left(x\right)+\frac{1}{2}\ln\left|\sec\left(x\right)+\tan\left(x\right)\right|+C_0$
