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# P3 Integration watch

1. A bit of help would be appreciated for the following:

1) Show that INT1 and 0) x/(1 + x)^1/2 = (2/3)(2 - root2)

2) INT: sec^(2)x/(1 + tan x)^3

3) INT: (sin x + 2 cos x)^2

For number three, the answer is (5/2)x + (3/4)sin2x - cos 2x. I get the same answer, apart from the - cos 2x bit. I multiplied out the bracket, so the 2nd term should be 4 sin x cos x. Then when I integrated this, I get either 4 sin^(2)x or 4 cos^(2)x. Could somewhere tell me how I correctly integrate this please?

Cheers everyone
2. (Original post by foowise)
A bit of help would be appreciated for the following:

1) Show that INT1 and 0) x/(1 + x)^1/2 = (2/3)(2 - root2)

2) INT: sec^(2)x/(1 + tan x)^3

3) INT: (sin x + 2 cos x)^2

For number three, the answer is (5/2)x + (3/4)sin2x - cos 2x. I get the same answer, apart from the - cos 2x bit. I multiplied out the bracket, so the 2nd term should be 4 sin x cos x. Then when I integrated this, I get either 4 sin^(2)x or 4 cos^(2)x. Could somewhere tell me how I correctly integrate this please?

Cheers everyone
For the last bit you were asking about, change it into a 2sin2x. And integrate that, to -cos2x
3. (Original post by foowise)
A bit of help would be appreciated for the following:

1) Show that INT1 and 0) x/(1 + x)^1/2 = (2/3)(2 - root2)
Use the substitution u²= 1+x
2u.du/dx = 1
dx = 2u.du

New limits: x=0, u=1...x=1, u=√2

=INT (√2 and 1) (u²-1 / u) . 2udu

=2 [INT (u²-1) du]

= 2 [u³/3 - u] (limits: √2 and 1)

= 2 [(2√2)/3 - √2] - [(1/3) - 1]

simplifies to (2/3)(2-√2)
4. Cheers guys, any chance of help for number 2?

And, just one more thing, could someone show that 4 cos 2x/sin^(2)2x = cosec^(2)x - sec^(2)x. I'm getting better at all those proofs, but this one has still got me. Cheers
5. (Original post by foowise)
Cheers guys, any chance of help for number 2?

And, just one more thing, could someone show that 4 cos 2x/sin^(2)2x = cosec^(2)x - sec^(2)x. I'm getting better at all those proofs, but this one has still got me. Cheers
2) ∫(sec^2(x))/((1+tanx)^3) dx

= ∫(1+tanx)'*(1+tanx)^3 dx

=((1+tanx)^4)/4 + k
6. (Original post by foowise)
Cheers guys, any chance of help for number 2?

And, just one more thing, could someone show that 4 cos 2x/sin^(2)2x = cosec^(2)x - sec^(2)x. I'm getting better at all those proofs, but this one has still got me. Cheers
cosec²(x) - sec²(x) = (cos²x-sin²x)/(sin²x*cos²x)

=(cos(2x))/(0.25*(2sinxcosx)²)

=4*cos2x/(sin2x)²

quod erat demonstrandum
7. (Original post by foowise)
Cheers guys, any chance of help for number 2?

And, just one more thing, could someone show that 4 cos 2x/sin^(2)2x = cosec^(2)x - sec^(2)x. I'm getting better at all those proofs, but this one has still got me. Cheers
L.H.S: = 4 cos 2x/sin^(2)2x

= [4 (cosx)^2 - (sinx)^2] / 4(sinx)^2(cosx)^2

= (cosx)^2 - (sinx)^2 / (sinx)^2(cosx)^2

= [(cosx)^2 / (sinx)^2(cosx)^2] - [(sinx)^2 / (sinx)^2(cosx)^2] <-----split up the fraction

= [1/(sinx)^2] - [1/(cosx)^2]

= (cosecx)^2 - (secx)^2

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