# Thread: Uni Yr1 first week exam - Integration questions

1. ## Uni Yr1 first week exam - Integration questions

HI, Uni starts next week and we have a first week paper to test our weaknesses. Integration has been a struggle at times but with the mock paper they have provided I am amiss with these 4 questions - sometimes not even understanding the actual question . Apologies for my notaion...

Any help would be greatly appreciated.
(i) Given that f & g are continous and that I = f(x) dx = 5 (limts 6 & 2) and I = g(x) dx = 9 (limits 6 & 2) calculate I = 4f(x) - g(x) dx (limits 2 & 6)

(ii) Using method by parts to express I = y^2*log(y) dy (limits x & 1) in the form
A(x) + I= B(y) dy (limits x & 1)

(iii) Integrate I = ((9x^2)/((1-x^3)^0.5)) by using substitution u = 1 - x^3

(iv) Find a suitable substitution of the form y = f(x) for I = x^3*e^(-x^2) dx (limits infinity & 0)

2. http://www.mathhelpforum.com/math-he...ial-19060.html
Why not learn to post in symbols? You can use LaTeX tags.
$$\int_6^2 {f(x)dx} = 5$$ gives $\displaystyle \int_6^2 {f(x)dx} = 5$.

3. Originally Posted by dojo
HI, Uni starts next week and we have a first week paper to test our weaknesses. Integration has been a struggle at times but with the mock paper they have provided I am amiss with these 4 questions - sometimes not even understanding the actual question . Apologies for my notaion...

Any help would be greatly appreciated.
(i) Given that f & g are continous and that I = f(x) dx = 5 (limts 6 & 2) and I = g(x) dx = 9 (limits 6 & 2) calculate I = 4f(x) - g(x) dx (limits 2 & 6)
So you are given that $\displaystyle \int_2^6 f(x)dx= 5$ and $\displaystyle \int_2^6 g(x) dx= 9$. You are asked for the value of $\displaystyle \int_2^6 4f(x)- g(x) dx$. To do that you need to know that "integration is linear"- that is that $\displaystyle \int af(x)+ bg(x) dx= a\int f(x)dx+ b\int g(x)dx$.

By the way, just saying that the limits are "6 & 2" doesn't tell us whether the integral is from 6 to 2" or vice-versa. That is, $\displaystyle \int_6^2$ or $\displaystyle \int_2^6$. I have assume the latter. Fortunately, for this problem, as long as all integrations are in the same order, it does not matter.

(ii) Using method by parts to express I = y^2*log(y) dy (limits x & 1) in the form
A(x) + I= B(y) dy (limits x & 1)
Apparently you are using "I= B(y)dy" to indicate $\displaystyle \int B(y)dy$- not very good notation- that "=" really doesn't belong there!
The integration by parts formula, which you are clearly expected to know, is
$\displaystyle \int u dv= uv- \int v du$.
Here, you have several different choices for "u" and "dv". Simplest is probably $\displaystyle u= ln(y)$ and $\displaystyle dv= y ^2 dy$

(iii) Integrate I = ((9x^2)/((1-x^3)^0.5)) by using substitution u = 1 - x^3
Okay, if $\displaystyle u= 1- x^3$ what is $\displaystyle du$?

(iv) Find a suitable substitution of the form y = f(x) for I = x^3*e^(-x^2) dx (limits infinity & 0)
Just at a wild guess you might try $\displaystyle u= x^2$!! what is du?