Final answer to the problem
Step-by-step Solution
How should I solve this problem?
- Integrate using tabular integration
- Integrate by partial fractions
- Integrate by substitution
- Integrate by parts
- Integrate by trigonometric substitution
- Weierstrass Substitution
- Integrate using trigonometric identities
- Integrate using basic integrals
- Product of Binomials with Common Term
- FOIL Method
- Load more...
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
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
Isolate $dx$ in the previous equation
Substituting $u$ and $dx$ in the integral and simplify
Take the constant $\frac{1}{7}$ out of the integral
Divide $1$ by $7$
Applying a reduction formula for the integral of the tangent function: $\displaystyle\int\tan(x)^{n}dx=\frac{1}{n-1}\tan(x)^{n-1}-\int\tan(x)^{n-2}dx$
Multiplying polynomials $\frac{1}{7}$ and $\frac{1}{11-1}\tan\left(u\right)^{10}-\int\tan\left(u\right)^{9}du$
Replace $u$ with the value that we assigned to it in the beginning: $7x$
The integral $-\frac{1}{7}\int\tan\left(u\right)^{9}du$ results in: $-\frac{1}{56}\tan\left(7x\right)^{8}+\frac{1}{42}\tan\left(7x\right)^{6}-\frac{1}{28}\tan\left(7x\right)^{4}+\frac{1}{14}\sec\left(7x\right)^2+\frac{1}{7}\ln\left(\cos\left(7x\right)\right)$
Gather the results of all integrals
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$