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# three maths questions! watch

1. Hey!

Was wondering if anyone could help me with this core 3 question:

Show that when you integrate [ e^x - e^(-x) ] between the limits of one and zero you get:

[(e - 1)^2]/e

And these two core 4 questions:

1. Integration of (e^x)/(1+e^2x) using x=lnu as a substitution

2. Integration of 1/ (1-x^2)^3/2 using substitution x=sinu

Thankyou
2. ∫e^x - e^(-x) dx

= [ex + e-x ] between 0,1

= e + 1/e - (1 + 1)

= (e² + 1 -2e)/e

= (e -1)²/e

1. Integration of (e^x)/(1+e^2x) using x=lnu as a substitution

∫(e^x)/(1+e^2x) dx

x = lnu

dx/du = 1/u

ex = u

=> ∫ u/(1 +u²)(1/u) du

= ∫ 1/(1 + u²) du.....I'll leave the rest, because it will get tedious and I assume you're allowed to quote this result anyway.

2. Integration of 1/ (1-x^2)^3/2 using substitution x=sinu

∫1/ (1-x^2)^3/2 dx

x=sinu

dx/du = cosu = √(1 - sin²u)

= (1 - x²)1/2

=> (1 - x²)1/2/(1 - x²)3/2

= (1 - x²)-1 => ∫sec²u du = tanu ... but what's this in terms of x?
3. (Original post by Looontag)
Hey!

Was wondering if anyone could help me with this core 3 question:

Show that when you integrate [ e^x - e^(-x) ] between the limits of one and zero you get:

[(e - 1)^2]/e

And these two core 4 questions:

1. Integration of (e^x)/(1+e^2x) using x=lnu as a substitution

2. Integration of 1/ (1-x^2)^3/2 using substitution x=sinu

Thankyou
∫ e^x - e^(-x) dx = [e^x +1/e^x] between 1 and 0
[e + 1/e] - [1+1]
e+1/e - 2 = e²/e + 1/e -2e/e = (e²-2e+1)/e = (e-1)²/e

i dont have the time to do the other ones because im revising economics
4. dammit.. posted too late

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