We can solve the integral $\int\tan\left(7x\right)^{11}dx$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it $u$), which when substituted makes the integral easier. We see that $7x$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part
$u=7x$
Intermediate steps
2
Now, in order to rewrite $dx$ in terms of $du$, we need to find the derivative of $u$. We need to calculate $du$, we can do that by deriving the equation above
$du=7dx$
3
Isolate $dx$ in the previous equation
$du=7dx$
4
Substituting $u$ and $dx$ in the integral and simplify
$\int\frac{\tan\left(u\right)^{11}}{7}du$
5
Take the constant $\frac{1}{7}$ out of the integral
$\frac{1}{7}\int\tan\left(u\right)^{11}du$
6
Divide $1$ by $7$
$\frac{1}{7}\int\tan\left(u\right)^{11}du$
7
Applying a reduction formula for the integral of the tangent function: $\displaystyle\int\tan(x)^{n}dx=\frac{1}{n-1}\tan(x)^{n-1}-\int\tan(x)^{n-2}dx$
The integral $-\frac{1}{7}\int\tan\left(u\right)^{9}du$ results in: $-\frac{1}{56}\tan\left(7x\right)^{8}+\frac{1}{42}\tan\left(7x\right)^{6}-\frac{1}{28}\tan\left(7x\right)^{4}+\frac{1}{14}\sec\left(7x\right)^2+\frac{1}{7}\ln\left(\cos\left(7x\right)\right)$
Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more