Find the indefinite integral. (Remember to use absolute values where appropri $$ \int \tan ^{5}\left(\frac{x}{10}\right) d x $$

Question

UpStudy AI · Accepted Answer

Calculate the integral $$ \int \tan^{5}\left(\frac{x}{10}\right) dx $$.
Evaluate the integral by following steps:
- step0: Evaluate using formulas and rules:
  $$\int \tan^{5}\left(\frac{x}{10}\right) dx$$
- step1: Evaluate the integral:
  $$\frac{\tan^{5-1}\left(\frac{1}{10}x\right)}{\frac{1}{10}\left(5-1\right)}-\int \tan^{5-2}\left(\frac{1}{10}x\right) dx$$
- step2: Subtract the numbers:
  $$\frac{\tan^{4}\left(\frac{1}{10}x\right)}{\frac{1}{10}\left(5-1\right)}-\int \tan^{5-2}\left(\frac{1}{10}x\right) dx$$
- step3: Subtract the numbers:
  $$\frac{\tan^{4}\left(\frac{1}{10}x\right)}{\frac{1}{10}\times 4}-\int \tan^{5-2}\left(\frac{1}{10}x\right) dx$$
- step4: Subtract the numbers:
  $$\frac{\tan^{4}\left(\frac{1}{10}x\right)}{\frac{1}{10}\times 4}-\int \tan^{3}\left(\frac{1}{10}x\right) dx$$
- step5: Multiply the numbers:
  $$\frac{\tan^{4}\left(\frac{1}{10}x\right)}{\frac{2}{5}}-\int \tan^{3}\left(\frac{1}{10}x\right) dx$$
- step6: Simplify the expression:
  $$\frac{5}{2}\tan^{4}\left(\frac{1}{10}x\right)-\int \tan^{3}\left(\frac{1}{10}x\right) dx$$
- step7: Evaluate the integral:
  $$\frac{5}{2}\tan^{4}\left(\frac{1}{10}x\right)-\left(5\tan^{2}\left(\frac{1}{10}x\right)-10\ln{\left(\left|\sec\left(\frac{1}{10}x\right)\right|\right)}\right)$$
- step8: Remove the parentheses:
  $$\frac{5}{2}\tan^{4}\left(\frac{1}{10}x\right)-5\tan^{2}\left(\frac{1}{10}x\right)+10\ln{\left(\left|\sec\left(\frac{1}{10}x\right)\right|\right)}$$
- step9: Add the constant of integral C:
  $$\frac{5}{2}\tan^{4}\left(\frac{1}{10}x\right)-5\tan^{2}\left(\frac{1}{10}x\right)+10\ln{\left(\left|\sec\left(\frac{1}{10}x\right)\right|\right)} + C, C \in \mathbb{R}$$
The indefinite integral of $$ \tan^{5}\left(\frac{x}{10}\right) $$ with respect to $$ x $$ is given by:

$$ \frac{5}{2}\tan^{4}\left(\frac{x}{10}\right) - 5\tan^{2}\left(\frac{x}{10}\right) + 10\ln\left(\left|\sec\left(\frac{x}{10}\right)\right|\right) + C $$

where $$ C $$ is an arbitrary constant.
The indefinite integral of $$ \tan^{5}\left(\frac{x}{10}\right) $$ with respect to $$ x $$ is: $$ \frac{5}{2}\tan^{4}\left(\frac{x}{10}\right) - 5\tan^{2}\left(\frac{x}{10}\right) + 10\ln\left(\left|\sec\left(\frac{x}{10}\right)\right|\right) + C $$ where $$ C $$ is a constant.

Find the indefinite integral. (Remember to use absolute values where appropri

Upstudy AI Solution

Answer

Solution

Beyond the Answer

Related Questions

Latest Calculus Questions