Final answer to the problem
$\frac{1}{2}\tan\left(x\right)\sec\left(x\right)+\frac{1}{2}\ln\left|\sec\left(x\right)+\tan\left(x\right)\right|+C_0$
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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
Taking the derivative of secant function: $\frac{d}{dx}\left(\sec(x)\right)=\sec(x)\cdot\tan(x)\cdot D_x(x)$
$\frac{d}{dx}\left(x\right)\sec\left(x\right)\tan\left(x\right)$
The derivative of the linear function is equal to $1$
$\sec\left(x\right)\tan\left(x\right)$
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
When multiplying two powers that have the same base ($\tan\left(x\right)$), you can add the exponents
$\tan\left(x\right)\sec\left(x\right)-\int\sec\left(x\right)\tan\left(x\right)^2dx$
7
Now replace the values of $u$, $du$ and $v$ in the last formula
$\tan\left(x\right)\sec\left(x\right)-\int\sec\left(x\right)\tan\left(x\right)^2dx$
Explain this step further
Intermediate steps
Applying the trigonometric identity: $\tan\left(\theta \right)^2 = \sec\left(\theta \right)^2-1$
$\tan\left(x\right)\sec\left(x\right)-\int\sec\left(x\right)\left(\sec\left(x\right)^2-1\right)dx$
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
$\tan\left(x\right)\sec\left(x\right)-\int\sec\left(x\right)\left(\sec\left(x\right)^2-1\right)dx$
Explain this step further
Intermediate steps
Multiply the single term $\sec\left(x\right)$ by each term of the polynomial $\left(\sec\left(x\right)^2-1\right)$
$\int\left(\sec\left(x\right)^2\sec\left(x\right)-\sec\left(x\right)\right)$
When multiplying exponents with same base you can add the exponents: $\sec\left(x\right)^2\sec\left(x\right)$
$\tan\left(x\right)\sec\left(x\right)-\int\left(\sec\left(x\right)^{3}-\sec\left(x\right)\right)dx$
9
Multiply the single term $\sec\left(x\right)$ by each term of the polynomial $\left(\sec\left(x\right)^2-1\right)$
$\tan\left(x\right)\sec\left(x\right)-\int\left(\sec\left(x\right)^{3}-\sec\left(x\right)\right)dx$
Explain this step further
Intermediate steps
Expand the integral $\int\left(\sec\left(x\right)^{3}-\sec\left(x\right)\right)dx$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately
$\tan\left(x\right)\sec\left(x\right)-\int\sec\left(x\right)^{3}dx+1\int\sec\left(x\right)dx$
Any expression multiplied by $1$ is equal to itself
$\tan\left(x\right)\sec\left(x\right)-\int\sec\left(x\right)^{3}dx+\int\sec\left(x\right)dx$
10
Simplify the expression
$\tan\left(x\right)\sec\left(x\right)-\int\sec\left(x\right)^{3}dx+\int\sec\left(x\right)dx$
Explain this step further
Intermediate steps
The integral of the secant function is given by the following formula, $\displaystyle\int\sec(x)dx=\ln\left|\sec(x)+\tan(x)\right|$
$\ln\left|\sec\left(x\right)+\tan\left(x\right)\right|$
11
The integral $\int\sec\left(x\right)dx$ results in: $\ln\left(\sec\left(x\right)+\tan\left(x\right)\right)$
$\ln\left(\sec\left(x\right)+\tan\left(x\right)\right)$
Explain this step further
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
Multiply the single term $\frac{1}{2}$ by each term of the polynomial $\left(\tan\left(x\right)\sec\left(x\right)+\ln\left(\sec\left(x\right)+\tan\left(x\right)\right)\right)$
$\frac{1}{2}\tan\left(x\right)\sec\left(x\right)+\frac{1}{2}\ln\left(\sec\left(x\right)+\tan\left(x\right)\right)+C_0$
$\frac{1}{2}\tan\left(x\right)\sec\left(x\right)+\frac{1}{2}\ln\left|\sec\left(x\right)+\tan\left(x\right)\right|+C_0$
Explain this step further
Final answer to the problem $\frac{1}{2}\tan\left(x\right)\sec\left(x\right)+\frac{1}{2}\ln\left|\sec\left(x\right)+\tan\left(x\right)\right|+C_0$
