Answer:
y=2/3x+4.3
Step-by-step explanation:
y=mx+c
(5-3)/(1--2)
2/3= gradient
y=2/3x +c
5=2/3(1)+c
-2/3
4.3=c
Answer:11/50
Step-by-step explanation:
Answer: could I see the equtions pls
Step-by-step explanation:
-2x+10^5 this should be I guess
Answer:
Step-by-step explanation:
.
It is apparently obvious we could expand the bracket and integrate term-by-term. This method would work but is very time consuming (and you could easily make a mistake) so we use a different method: integration by substitution.
Integration by substitution involves swapping the variable
for another variable which depends on x:
. (We are going to choose
for this question).
The very first step is to choose a suitable substitution. That is, an equation
which is going to make the integration easier. There is a trick for spotting this however: if an integral contains both a term and it's derivative then use the substitution
.
Your integral contains the term
. The derivative is
and (ignoring the constants) we see
is also in the integral and so the substitution
will unravel this integral!
Step 2: We must now swap the variable of integration from x to u. That means interchanging all the x's in the integrand (the term being integrated) for u's and also swapping (dx" to "du").
![u=x^2+3 \Rightarrow \frac{du}{dx}=2x \Rightarrow dx = \frac{1}{2x} du](https://tex.z-dn.net/?f=u%3Dx%5E2%2B3%20%5CRightarrow%20%5Cfrac%7Bdu%7D%7Bdx%7D%3D2x%20%5CRightarrow%20dx%20%3D%20%5Cfrac%7B1%7D%7B2x%7D%20du)
Then,
.
The substitution has made this integral is easy to solve!
![\int \frac{3}{2}u^4\ du= \frac{3}{10}u^5 + C](https://tex.z-dn.net/?f=%5Cint%20%5Cfrac%7B3%7D%7B2%7Du%5E4%5C%20du%3D%20%5Cfrac%7B3%7D%7B10%7Du%5E5%20%2B%20C)
Finally we can substitute back to get the answer in terms of x:
![\int 3x(x^2+3)^4 \ dx = \frac{3}{10}(x^2+3)^5+C](https://tex.z-dn.net/?f=%5Cint%203x%28x%5E2%2B3%29%5E4%20%5C%20dx%20%3D%20%5Cfrac%7B3%7D%7B10%7D%28x%5E2%2B3%29%5E5%2BC)