Could someone work this question out so I understand it.
Find the indefinite intregral
(lnx)^7 / x dx
Use C as the arbitrary constant.
The best method here is integration by parts. Most of the times when you see products of functions AND the product of one with the others derivative gives a simple to integrate function.
Do you see that [ln(x)]' = 1/x. And when you multiply it with a power of x you get a power of x?
The standard form of the integration by parts is:
integral_{f'(x)*g(x)}dx = f(x)*g(x) - integral_{f(x)*g'(x)}dx
The idea is to pick f and g so it'll all be simpler. We already picked out g(x) = ln(x)
Now we need f'(x). This can only be x^7. But what functions derivative is x^7? We have to to the opposite of derivation ---> integration!
f(x) = integral_{f'(x)}dx = integral_{x^7}dx = (x^8)/8.
Just check! [(x^8)/8]' = [(1/8)*(x^8)]' = (1/8)*8*(x^7) = x^7
integral_{f'(x)*g(x)}dx = integral_{(x^7)*ln(x)}dx = integral_{ [(x^8)/8]' * ln(x) }dx =
= [(x^8)/8] * ln(x) - integral_{ [(x^8)/8] * ln'(x) }dx =
= [(x^8)/8] * ln(x) - integral_{ [(1/8)*(x^8)] * 1/x }dx =
= [(x^8)/8] * ln(x) - (1/8)* integral_{ [x^(8-1)] }dx =
= [(x^8)/8] * ln(x) - (1/8)* integral_{ [x^7] }dx =
= [(x^8)/8] * ln(x) - (1/8)*(x^8)/8 + real_constant =
= [(x^8)/8] * ln(x) - (x^8)/64 + real_constant = [(x^8)/8] * [ln(x) - (1/8)] + real_constant
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . = [(x^8)/64] * [ 8*ln(x) - 1 ] + real_constant
Sometimes there are different ways to solve a problem, here is another:
∫(lnx)^7 / x dx
We note that (lnx)' = 1/x
and substitute y=ln(x), dy=dx/x
∫(lnx)^7 / x dx
= ∫ y^7 dy
= y^8/8 + C
= ln(x)^8/8 + C
To find the indefinite integral of (lnx)^7 / x dx, you can use a technique called integration by parts. Integration by parts is based on the formula:
∫u dv = uv - ∫v du
In this case, let's choose u = (lnx)^7 and dv = dx. This way, we can differentiate u to find du, and integrate dv to find v.
Step 1: Finding du
Differentiating u = (lnx)^7 with respect to x:
du = 7(lnx)^6 * (1/x) * dx
Step 2: Finding v
Integrating dv = dx:
v = ∫dx = x
Step 3: Applying the integration by parts formula
Now, we can substitute the values we found into the integration by parts formula:
∫u dv = uv - ∫v du
∫ (lnx)^7 / x dx = (lnx)^7 * x - ∫x * [7(lnx)^6 * (1/x) * dx]
Simplifying further, we get:
∫ (lnx)^7 / x dx = (lnx)^7 * x - 7 * ∫(lnx)^6 dx
Step 4: Simplifying the expression
The integral ∫(lnx)^6 dx can be further solved using the substitution method. Let's substitute t = lnx, which means dt = (1/x) dx.
So, the expression becomes:
∫(lnx)^7 / x dx = (lnx)^7 * x - 7 * ∫(lnx)^6 dx
= (lnx)^7 * x - 7 * ∫t^6 dt (using the substitution t = lnx)
= (lnx)^7 * x - 7 * (t^7 / 7) + C (integrate t^6 with respect to t)
= (lnx)^7 * x - (lnx)^7 + C (simplification)
Therefore, the indefinite integral of (lnx)^7 / x dx is:
∫ (lnx)^7 / x dx = (lnx)^7 * x - (lnx)^7 + C
where C is the arbitrary constant.