Integrate sec^2x tan^4x: Tan^5(x)/5

Thread starter bard
Start date Dec 8, 2003
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Integration

In summary, the formula for integrating sec^2x tan^4x is tan^5(x)/5 + C. You can use either the substitution method or the integration by parts method to integrate this expression. The steps for using substitution are to substitute u = tan(x), rewrite the integral, integrate, and then substitute back the original variables. The integration by parts method can be used when the integral cannot be simplified using substitution. An example of using this method is shown, where the integral is broken down into smaller parts until it can be solved, and then the original variables are substituted back to get the final answer.

Dec 8, 2003

bard

hey everyone i need help with this antiderivative

[tex]\int\sec^{2}x\tan^{4}x[/tex]

my guess is that it is tan^5(x)/5

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Dec 8, 2003

gnome

Good guess!

Dec 9, 2003

KLscilevothma

Don't forget the constant of integration.

1. What is the formula for integrating sec^2x tan^4x?

The formula for integrating sec^2x tan^4x is tan^5(x)/5 + C.

2. How do you integrate sec^2x tan^4x?

To integrate sec^2x tan^4x, you can use the substitution method or the integration by parts method.

3. Can you explain the steps for integrating sec^2x tan^4x using substitution?

Yes, the steps for integrating sec^2x tan^4x using substitution are:
1. Substitute u = tan(x), which means du = sec^2x dx
2. Rewrite the integral as ∫u^4 du
3. Integrate ∫u^4 du to get u^5/5
4. Substitute back u = tan(x) to get tan^5(x)/5 + C as the final answer.

4. When do you use the integration by parts method to integrate sec^2x tan^4x?

You can use the integration by parts method to integrate sec^2x tan^4x when the integral cannot be simplified using the substitution method.

5. Can you provide an example of using the integration by parts method to integrate sec^2x tan^4x?

Sure, an example of using the integration by parts method to integrate sec^2x tan^4x is:
∫sec^2x tan^4x dx
= ∫sec^2x tan^2x * tan^2x dx
= tan^3x/3 * tan^2x - ∫tan^3x/3 * 2tanx sec^2x dx
= tan^3x/3 * tan^2x - 2/3 * ∫tan^4x sec^2x dx
You can continue this process until the integral becomes solvable, and then substitute back the original variables to get the final answer.