P3 Integration

A bit of help would be appreciated for the following:

1) Show that INT :frown:

1 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

Reply 1

fishpaste

foowise

A bit of help would be appreciated for the following:

1) Show that INT :frown:

1 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

Reply 2

m:)ckel

14

foowise

A bit of help would be appreciated for the following:

1) Show that INT :frown:

1 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)

Reply 3

foowise

OP

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

Reply 4

zizero

1

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

Reply 5

zizero

1

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

Reply 6

m:)ckel

14

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

P3 Integration

Related discussions

Latest

Trending

Trending

Articles for you