Not sure if it is right, I have check with the answer in the book and a few integral calculators but they seem to get a different answer
∫ sec^3(x)tan^3(x) dx
∫ sec^3(x)tan(x)(sec^2(x)-1) dx
∫ tan(x)sec(x)[sec^4(x)-sec^2(x)] dx
∫ tan(x)sec(x)[(tan^2(x)+1)^2-tan^2(x)-1] dx
∫ tan(x)sec(x)(1+2tan(x)+tan^2(x)-tan^2(x)-1) dx
∫ 2tan^2(x)sec(x) dx
∫ 2(sec^2(x)-1)sec(x) dx
∫ 2sec^3(x)-2sec(x) dx
2∫ sec^3(x) dx -2∫ sec(x) dx
*reduction formula
2tan(x)sec(x)-ln(tan(x)+sec(x))
In google type:
wolfram alpha
When you see lis of results click on:
Wolfram Alpha:Computational Knowledge Engine
When page be open in rectangle type:
integrate sec^3(x)tan^3(x) dx
and click option =
After few secons you will see result.
Then click option Show steps
I did, I got a completely different result. Also, SAY SOMETHING USEFUL NEXT TIME!!! tired of you spamming that stupid reply
Try to simplify your "different answers".
Probably your "different answers" is same solutions write in different form.
right, because my answer can be simplified into:
sec^5(x)/5-sec^3(x)/3
To solve the integral ∫ sec^3(x)tan^3(x) dx, you have followed the correct steps until the last line. Let's review the process together:
1. Start with ∫ sec^3(x)tan^3(x) dx.
2. Rewrite tan^3(x) as tan(x)[sec^2(x) - 1] to obtain ∫ sec^3(x)tan(x)(sec^2(x) - 1) dx.
3. Further simplify by replacing sec^2(x) - 1 with tan^2(x) to get ∫ tan(x)sec(x)[tan^2(x) + 1 - tan^2(x)] dx.
4. Combine like terms within the brackets to obtain ∫ tan(x)sec(x)(1 + 2tan^2(x)) dx.
5. Distribute tan(x)sec(x) and simplify to get ∫ 2tan^2(x)sec^3(x) dx.
6. Apply a reduction formula to integrate sec^3(x) dx separately. This formula states that ∫ sec^n(x) dx = (1/(n - 1)) sec^(n-2)(x)tan(x) + (n - 2)/(n - 1) ∫ sec^(n-2)(x) dx.
7. Apply the reduction formula and integrate the first term, resulting in 2tan(x)sec^3(x) - 2∫ sec(x) dx.
8. Finally, integrate the second term using the integral of sec(x), which is ln|sec(x) + tan(x)|.
Combining the results from step 7 and step 8, you correctly obtained the final answer 2tan(x)sec(x) - ln|sec(x) + tan(x)| + C, where C represents the constant of integration.
If the answer you obtained differs from the one in the book or integral calculators, it is possible that a sign error or minor mistake occurred during the calculations. Double-check your steps, review the algebraic simplifications, and revisit any trigonometric identities or formulas you applied.