Integration

[squance]

Integrate the following:

sin^2(x) dx
(I got cos^3 (x)/3 - x + c as an answer but the answer says cos^3 (x)/ 3 - cos (x) + C. Why is it so??)

(x^4+1) loge(x^5+ 5x) dx
(I have no idea how to work this out!!!)

[ice_blockie]

Part 2: You have to use the substitution method, which is the 'opposite' of a chain rule for differentiation.

i.e.

[squance]

Part 2: You have to use the substitution method.

Here's what i did:
(x^4+1) loge(x^5+ 5x) dx

let u = x^5 + 5x
du/dx = 5x^4 + 5
du/dx = 5(x^4+1)
du/5dx = x^4 + 1

integration....
du/5dx logeu dx
1/5logeu du

Then what next??? You can't integrate log, can you???

[cara.mel]

You can use integration by parts I guess



Then all you have to do is differentiate log. (set it to the f(x) half, so f(x) = log(x^5+5x), g'(x) = (x^4+1)

Anyway I am no good at maths, I havent bothered to check if what I said would be helpful but that what I've been taught to do in case of PANIC, LOG

[AppleXY]

For Part 1, you gotta use the trig identities

i.e. =

[ice_blockie]

Hey squance, is the answer to part 2

[dcc]

Let






Now from here on, you have to experiment with a technique called 'integration by parts' (as caramel said), but a simple way of using this technique is to consider:







Therefore using this integral as above: