1
Solved example of Trigonometric integrals
$\int\left(\sec\left(x\right)^6-\sec\left(x\right)^4\right)dx$
2
The integral of a sum of two or more functions is equal to the sum of their integrals
$\int\sec\left(x\right)^6dx+\int-\sec\left(x\right)^4dx$
3
Take the constant out of the integral
$\int\sec\left(x\right)^6dx-\int\sec\left(x\right)^4dx$
4
Simplify the integral of secant applying the reduction formula, $\displaystyle\int\sec(x)^{n}dx=\frac{\sin(x)\sec(x)^{n-1}}{n-1}+\frac{n-2}{n-1}\int\sec(x)^{n-2}dx$
$\frac{\sec\left(x\right)^{5}\sin\left(x\right)}{5}+\frac{6-2}{6-1}\int\sec\left(x\right)^{\left(6-2\right)}dx-\int\sec\left(x\right)^4dx$
5
Subtract the values $6$ and $-2$
$\frac{\sec\left(x\right)^{5}\sin\left(x\right)}{5}+\frac{4}{5}\int\sec\left(x\right)^{4}dx-\int\sec\left(x\right)^4dx$
$\frac{\sec\left(x\right)^{5}\sin\left(x\right)}{5}+\frac{4}{5}\int\sec\left(x\right)^{4}dx-\int\sec\left(x\right)^4dx$
7
Adding $\frac{4}{5}\int\sec\left(x\right)^{4}dx$ and $-1\int\sec\left(x\right)^{4}dx$
$\frac{\sec\left(x\right)^{5}\sin\left(x\right)}{5}-\frac{1}{5}\int\sec\left(x\right)^{4}dx$
8
Simplify the integral of secant applying the reduction formula, $\displaystyle\int\sec(x)^{n}dx=\frac{\sin(x)\sec(x)^{n-1}}{n-1}+\frac{n-2}{n-1}\int\sec(x)^{n-2}dx$
$\frac{\sec\left(x\right)^{5}\sin\left(x\right)}{5}-\frac{1}{5}\left(\frac{\sec\left(x\right)^{3}\sin\left(x\right)}{3}+\frac{2}{3}\int\sec\left(x\right)^{2}dx\right)$
9
The integral of $\sec(x)^2$ is $\tan(x)$
$\frac{\sec\left(x\right)^{5}\sin\left(x\right)}{5}-\frac{1}{5}\left(\frac{\sec\left(x\right)^{3}\sin\left(x\right)}{3}+\frac{2}{3}\tan\left(x\right)\right)$
10
As the integral that we are solving is an indefinite integral, when we finish we must add the constant of integration
$\frac{\sec\left(x\right)^{5}\sin\left(x\right)}{5}-\frac{1}{5}\left(\frac{\sec\left(x\right)^{3}\sin\left(x\right)}{3}+\frac{2}{3}\tan\left(x\right)\right)+C_0$