- #1
bard
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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
[tex]\int\sec^{2}x\tan^{4}x[/tex]
my guess is that it is tan^5(x)/5
The formula for integrating sec^2x tan^4x is tan^5(x)/5 + C.
To integrate sec^2x tan^4x, you can use the substitution method or the integration by parts method.
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.
You can use the integration by parts method to integrate sec^2x tan^4x when the integral cannot be simplified using the substitution method.
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.