Calculus integration question - substitution method?

Question

The question says to evaluate the indefinite integral by means of substitution. How do I do this?

(integration S symbol thingy) [ x( x^2 + 4 )^7 ]dx

Thanks for the help in advance!

Answer 1

this is a "u-substiution" type question. U-substiution involves selecting a portion of the thing you are integrating and setting that equal to a variable (usually "u") Then you take the derivative of what you are attempting to substitute out of the equation (what "u" equals) and set that equal to "du". Then you somehow find a way to substiute in for "du". After that, if you picked the proper values for "u" and "du", you should be able to integrate. Afterwards, sub your original expression for "u" back into the question to get a final answer.
In this problem, u = x^2 +4 so du = 2x dx
in order to get the expression 2x dx, convert your original expression to S[1/2 *2x( x^2 + 4 )^7 ]dx.
Sub for "u": S[1/2 *2x(u)^7] dx
Sub for "du": S[1/2 (u)^7du]
Integrate to get: 1/2 * 1/8 (u)^8 +C
sub back in: 1/2*1/8*(x^2 +4)^8 + C
simplify: 1/16 (x^2 +4)^8 + c

Answer 2

Yup, substitution. If x^2 + 4 = u, then then du = 2x dx. We then divide by 2, giving du/2 = x dx.

Then we can subsitute u^7 / 2 du, and intigrate normally, giving u^8 / 16. Replacing u as x^2 +4, we get (x^2 + 4)^8 / 16 + C.

Answer 3

First of we will use the rule that xdx = 1/2d(x^2), we'll get:
1/2 INT ((x^2 + 4)^7d(x^2))
Then well add number 4, which is a constant, to x^2:
1/2 INT ((x^2 + 4)^7d(x^2+4))
This will be equal to:
1/2 (x^2 + 4)^8/8 = (x^2 + 4)^8/16

Answer 4

Let u = x² + 4
du = 2x.dx
x.dx = du / 2
I = (1/2) ∫ u^7 du
I = (1/2).u^8 / 8 + C
I = (1/16).(x² + 4)^8 + C

Answer 5

i have no idea. im in algebra one right now. sorry