Question 1208301: Hi can you help please:
int (3 x^3) (sqrt(16-x^2)) dx
Just not sure if my trig substitutions are working Found 3 solutions by Edwin McCravy, math_tutor2020, mccravyedwin:Answer by Edwin McCravy(20056) (Show Source):
That x3 factor is going to make trig substitution difficult.
So let's not try trig substitution, but try algebraic substitution instead.
=>
Your teacher may let you leave it like that, since the
calculus part is over, and the rest is algebra. But
here's the remaining algebra part:
factor out
Edwin
The rule we'll use specifically is
This is in the blue box underneath example 4 on that web link I posted.
Notice this rule corresponds to the template
In this case a = 4 and b = 1.
Let's call this equation (1) to use later.
Then, Let's call this equation (2) to use later.
Also Let's call this equation (3) to use later.
Replace any mention of with , but there won't be a dot between the d and theta. I couldn't get it to format properly.
Let's apply those items into the integral we want to evaluate.
= Each red box represents a thing we'll replace in the next line.
= Substitute in equations (1), (2), and (3).
=
=
=
=
=
= Let u = cos(theta), which means du = -sin(theta)*dtheta and sin(theta)*dtheta = -du
=
= Don't forget about the plus C. I've seen many students make this error.
=
From here you'll need to replace each "u" with an expression of x.
Recall that and
Which would mean and
I'll let the student take over from here.
You should arrive at the final answer of or some equivalent variation of this.
You can use various online calculators (eg: WolframAlpha or GeoGebra) to confirm the answer.
Here's what it looks like when using WolframAlpha https://www.wolframalpha.com/input?i=int%28+3x%5E3*sqrt%2816-x%5E2%29%2C+dx+%29
The input typed in was int( 3x^3*sqrt(16-x^2), dx ) or you could do integral( 3x^3*sqrt(16-x^2), dx )
Here is the way to solve it by trig substitution. It isn't so bad after all.
Draw a right tringle with angle θ, hypotenuse 4, opposite leg x, and
adjacent leg . As a convention we always put x on the opposite
leg [although it wouldn't matter if we used cofunctions instead.]
[I can't write dθ using the program on the site, so I'll use "dO" for "dθ", so don't be upset.
Use the substitutions above, taken straight from the right triangle:
Take out all the constants:
When you have an odd power of a sine or cosine, save a sine or cosine
for the derivative and replace the even power that's left with an
identity. so we break up sin3θ = (sin2θ)sinθ = (1-cos2θ)sinθ
We need to insert a - sign since the derivative of cosine is negative sine:
Factor out 1/240
That is the same answer as I got above using algebraic substitution.
Edwin
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