Math Help - integrationand differeniation help

1. integrationand differeniation help

$\int_1^4 g(x) dx = 5$
and $\int_2^4 g(x) = 2$

evulate (i) $\int_4^1 g(x) - \sqrt{x} dx$

and (ii) $\int_1^2 g(x) dx$

And also another qn. $\frac{d}{dx} ln \sin x$
thanks for the help.

2. Originally Posted by helloying
$\int_1^4 g(x) dx = 5$
and $\int_2^4 g(x) = 2$

evulate (i) $\int_4^1 g(x) - \sqrt{x} dx$

and (ii) $\int_1^2 g(x) dx$

And also another qn. $\frac{d}{dx} ln \sin x$
thanks for the help.
(i) Using some simple properties of the integral, you have $\int_4^1 g(x) - \sqrt{x} dx = - \int_1^4 g(x) \, dx + \int_1^4 x^{1/2} \, dx$.

(ii) Note that $\int_1^2 g(x) \, dx = \int_1^4 g(x) \, dx - \int_2^4 g(x) \, dx$.

For the final question, use the chain rule.

3. Originally Posted by mr fantastic
(i) Using some simple properties of the integral, you have $\int_4^1 g(x) - \sqrt{x} dx = - \int_1^4 g(x) \, dx + \int_1^4 x^{1/2} \, dx$.

(ii) Note that $\int_1^2 g(x) \, dx = \int_1^4 g(x) \, dx - \int_2^4 g(x) \, dx$.

For the final question, use the chain rule.

thank you. but for the last qn, i dont know what is a chain rule. how i would attemp this qn would be: because $\frac{d}{dx}ln x = 1/x$, then the ans for the qn will be 1/sinx . but i check the book my ans is wrong.

4. Originally Posted by helloying
thank you. but for the last qn, i dont know what is a chain rule. how i would attemp this qn would be: because $\frac{d}{dx}ln x = 1/x$, then the ans for the qn will be 1/sinx . but i check the book my ans is wrong.
I'm not sure why you would be attempting this question if you don't know the chain rule.

Chain rule: $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$.

$y = \ln \sin x$.

Let $u = \sin x$.

Then $y = \ln u \Rightarrow \frac{dy}{du} = \frac{1}{u} = \frac{1}{\sin x}$ and $\frac{du}{dx} = \cos x$.