Intergrate ¡ì sec^3(x) dx
could anybody please check this answer. are the steps correct? thanks.
= ¡ì sec x d tan x
= sec x tan x - ¡ì tan x d sec x
= sec x tan x - ¡ì sec x tan^2(x) dx
= sec x tan x + ¡ì sec x dx - ¡ì sec^3(x) dx
= sec x tan x + ln |sec x + tan x| - ¡ì sec^3(x) dx
=¡ì sec^3(x) dx = (1/2)(sec x tan x + ln |sec x + tan x|) + C1
¡ì [3x sin x/cos^4(x)] dx
= -3 ¡ì [x/cos^4(x)] d cos x
= ¡ì x d sec^3(x)
= x sec^3(x) - ¡ì sec^3(x) dx
= x sec^3(x) - (1/2) sec x tan x - (1/2) ln |sec x + tan x| + C2
I'm not sure if your integration is correct or not, not all of your symbols converted to ASCII. I plugged sec3(x) into a piece of software and got an answer that looks slightly different from yours, but I'm not positive. If you still need help with this post a new question so it's easy to find.
s=integral
let i=S(sec^3x)dx
i=S(sec^3x)=S(sec^2x.secx).dx
=tanxsecx-S(tan^2x.secx)dx=
tanxsecx-S(sec^2x-1)secx.dx=
tanxsecx-S(sec^3x)dx+S(secx)dx
2i=tanxsecx+ln|secx+tanx|+x
i=(tanxsecx+ln|secx+tanx|+c)(0.5)
-cos(2x)upon2
To integrate ∫ sec^3(x) dx, you can use a technique called integration by parts. Here are the steps to solve it:
Step 1: Rewrite sec^3(x) as sec(x) * sec^2(x).
Step 2: Apply integration by parts, where u = sec(x) and dv = sec^2(x) dx.
Step 3: Calculate du/dx and v, which are du = sec(x) * tan(x) dx and v = tan(x).
Step 4: Use the formula for integration by parts: ∫ u dv = u*v - ∫ v du.
Applying the integration by parts formula, we have:
∫ sec^3(x) dx = sec(x) * tan(x) - ∫ tan(x) * sec(x) * tan(x) dx
Step 5: Simplify the integral on the right-hand side.
∫ tan(x) * sec(x) * tan(x) dx = ∫ tan^2(x) sec(x) dx
Step 6: Rewrite tan^2(x) as sec^2(x) - 1.
∫ (sec^2(x) - 1) * sec(x) dx
Step 7: Distribute and integrate each term separately.
= ∫ sec^3(x) dx - ∫ sec(x) dx
Step 8: Solve the first integral using equation from Step 7.
∫ sec^3(x) dx = ∫ sec^3(x) dx - ∫ sec(x) dx
Rearranging the equation, we get:
2 ∫ sec^3(x) dx = ∫ sec(x) dx
∫ sec^3(x) dx = (1/2) ∫ sec(x) dx
Step 9: Integrate ∫ sec(x) dx using the natural logarithm identity.
= (1/2) ln|sec(x) + tan(x)|
Step 10: Substitute back into the original equation to get the final result.
∫ sec^3(x) dx = (1/2) ln|sec(x) + tan(x)| + C
So, the correct answer is (1/2) ln|sec(x) + tan(x)| + C, where C is the constant of integration.