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How do you find the integral of #tanh^3x dx#?

Answer 1

Isabella Ackerman

#inttanh(x)^3dx=ln(|cosh(x)|)+1/2sech(x)^2+C#, #C in RR#

#I=inttanh(x)^3dx#

#=intsinh(x)^3/(cosh(x)^3)dx#

Because #sinh(x)^2=cosh(x)^2-1#,

#I=int(sinh(x)(cosh(x)^2-1))/(cosh(x)^3)dx#

Now let #u=cosh(x)#

#du=sinh(x)dx#

So:

#I=int(u^2-1)/u^3du#

#=int1/udu-int1/u^3du#

#=ln(|u|)+1/(2u^2)+C#, #C in RR#

#=ln(|cosh(x)|)+1/2sech(x)^2+C#, #C in RR#

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Answer 2

Evelyn Agard

To find the integral of tanh^3(x) dx, you can use integration by parts method. Let u = tanh(x) and dv = tanh^2(x) dx. Then, differentiate u to get du, and integrate dv to get v. Once you have u, du, v, and dv, you can apply the integration by parts formula:

∫ tanh^3(x) dx = ∫ u dv = uv - ∫ v du

By substituting the values of u, du, v, and dv, you can calculate the integral. The result would be:

∫ tanh^3(x) dx = (tanh(x) * (tanh^2(x) - x))/2 + ln|cosh(x)|/2 + C

Where C is the constant of integration.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.