1. ## integration question

$\displaystyle \int_1^{7} f(x) dx = 19$ and $\displaystyle \int_5^{7} f(x)dx = 12$

then find
$\displaystyle \int_1^{5} 3f(x) dx$

$\displaystyle \int_1^{7} f(J) dJ$

how would i work this out? cheers

2. Any constant can be moved inside and outside of an integral freely. So:

$\displaystyle \int_1^{5} 3f(x) dx = 3 \int_{1}^{5}f(x)dx$

Now if we simply use the definition of integral bounds, we can construct these bounds by combining others.

$\displaystyle \int_{1}^{5}f(x)dx=\int_{1}^{7}f(x)dx-\int_{5}^{7}f(x)dx=19-12=7$

Multiply by 3 and I get an answer of 21.

3. i have depicted this problem for you visually to understand it.

4. Originally Posted by Jameson
Any constant can be moved inside and outside of an integral freely. So:

$\displaystyle \int_1^{5} 3f(x) dx = 3 \int_{1}^{5}f(x)dx$

Now if we simply use the definition of integral bounds, we can construct these bounds by combining others.

$\displaystyle \int_{1}^{5}f(x)dx=\int_{1}^{7}f(x)dx-\int_{5}^{7}f(x)dx=19-12=7$

Multiply by 3 and I get an answer of 21.
ahh okay, do you do the same thing for part (b)? $\displaystyle \int_1^7 f(J) dJ$ ?

EDIT: Sorry i understand now $\displaystyle \int_1^7 f(J) dJ = 19$

5. yes it is possible to change x with another variable like j and the intgral remain the same.