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# 𝘶-substitution: defining 𝘶 (more examples)

AP.CALC:
FUN‑6 (EU)
,
FUN‑6.D (LO)
,
FUN‑6.D.1 (EK)
A common challenge when performing 𝘶-substitution is to realize which part should be our 𝘶.

## Want to join the conversation?

• What would the answer be for both problems after you put your answer back in terms of x? Do you have to take the antiderivative of u^10 and then plug ln(x) back in for u and plug 1/x back in for du?
• Greetings
Here are the solutions to the two problems:

1) ∫ ((ln (x))^10)/x) dx= 1/11 ln (x)^11 + C

2) ∫ tan(x) dx = -ln |cos(x)| + C

I hope this helps!
• for the tan x question, can I instead plug in u= sin x and u' = cos x?
• Actually No, because the dx term wouldn't have been able to be replaced. here, let's see how it works out.

u = sin x
du/dx = cos(x)
du = cos(x) dx

Meanwhile in the problem you start with sin(x)/cos(x) dx. if you replace sin(x) with u you have:

u/cos(x) dx or u (1/cos(x))dx but du = cos(x) dx so you can't sub that out for du. Let me know if that doesn't make sense.
• huh? but my teacher gave me a list if integral formulas and said that integral of tanx =
log l sec(x)) l + c

i am not sure as to what i should believe... does Sal's answer somehow simplify down into this? and if so, how?
• They are equivalent! Following through with Sal's solution, we arrive at:
-log |cos 𝑥| + 𝐶
But recall that 𝑎log 𝑏 = log(𝑏ª). Hence:
-log |cos 𝑥| + 𝐶 = log |(cos 𝑥)⁻¹| + 𝐶
The reciprocal of cosine is secant, hence:
-log |cos 𝑥| + 𝐶 = log |(cos 𝑥)⁻¹| + 𝐶 = log |sec 𝑥| + 𝐶
As desired. Comment if you have questions!
• i tried to solve the second example in other form,

int(tan(x))dx
1/(sec(x))^2int(tan(x)(sec(x))^2 dx

u= tg(x)
du=(sec(x))^2dx

1/(sec(x))^2 int(u du)

1/((sec(x))^2 u^2/2 +c

(tg(x))^2/2(sec(x))^2 +c

(sen(x))^2/2 +c

did i made a mistake ??
• We can't move functions outside of the integral.

, ∫𝑓(𝑥)𝑑𝑥 = ∫𝑓(𝑥)𝑔(𝑥)∕𝑔(𝑥)𝑑𝑥 ≠ 1∕𝑔(𝑥)∙ ∫𝑓(𝑥)𝑔(𝑥)𝑑𝑥
• I wanna know if it is possible to simplify ∫ln(x)^10* dx to be 10∫ln(x) first before substituting. Because then the final answer would be 10 * u^2/2 = [5ln(x)^2 + C] instead of ln(x)^11/11 + C
• Look at the brackets carefully. It is (ln(x))^10 NOT ln(x^10). Theres a differene, we're raising whatever ln x is to the power 10 instead of raising x to the power 10 and then finding the resulting ln. Hence we cannot use that log property
(1 vote)
• Hey, Sal. How would you solve an integral if the integrand doesn't contain the derivative of the function, given that the derivative of said function is not a constant, but another function with variables? My teacher taught us that you can only multiply by constants to create the derivative.

Ex: Integral of 3 (sin x)^-2 dx

My first instinct was to set u = sin x. However, that would result in du = cos x dx, and you can't create that derivative by only introducing constants. My next thought was to set u equal to something else, but I don't know what.

Follow-up question: How do you know what to set u equal to in a question like this?
• What happens when the expression has a constant .... like, how to do this one indefinite integral of x/(x^2 + a^2)^3/2 ....??
Plz solve in detailed manner,
Thanx.
• Take u = a^2 + x^2.
du = 2x dx.
Write the numerator in terms of du, the denominator in terms of u, and apply the reverse power rule.
• Isn't the antiderivative of (1/x) actually ln(|x|) - with the absolute value?
(1 vote)
• The ln(x) must have an absolute value sign on it! ln(|x|)