Answer:
Explanation:
![\int 3x(x^2+3)^4 \ dx]()
.
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
![x]()
for another variable which depends on x:
![u(x)]()
. (We are going to choose
![u]()
for this question).
The very first step is to choose a suitable substitution. That is, an equation
![u=f(x)]()
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
![u=\text{The Term}]()
.
Your integral contains the term
![x^2 + 3]()
. The derivative is
![2x]()
and (ignoring the constants) we see
![x]()
is also in the integral and so the substitution
![u=x^2+3]()
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 (du)/(dx)=2x \Rightarrow dx = (1)/(2x) du]()
Then,
![\int 3x(x^2+3)^4 \ dx = \int 3x \cdot u^4 \cdot (1)/(2x) du = \int (3)/(2)u^4\ du]()
.
The substitution has made this integral is easy to solve!
![\int (3)/(2)u^4\ du= (3)/(10)u^5 + C]()
Finally we can substitute back to get the answer in terms of x:
![\int 3x(x^2+3)^4 \ dx = (3)/(10)(x^2+3)^5+C]()