Evaluate the integral by making the given substitution. (Use C for the constant of integration.)
∫ dt/(1-7)^7, u=1-7t
I suspect you mean
∫ dt/(1-7t)^7, u=1-7t
so if u = 1-7t
then du = -7 dt
or dt = -du/7
so we have
∫( -du/7)/u^7
or
-(1/7) ∫ u^-7 du
-(1/7)(-1/6) u^-6
(1/42) u^-6 + C
1/[42(1-7t)^6] + C
To evaluate the integral ∫ dt/(1-7t)^7 using the substitution u = 1 - 7t, we can first find the derivative of u with respect to t.
Taking the derivative of both sides of the equation u = 1 - 7t with respect to t, we get:
du/dt = -7
Now, we can solve for dt in terms of du by rearranging the equation:
dt = du / (-7)
Substituting u = 1 - 7t into the original integral, we have:
∫ dt/(1-7t)^7 = ∫ du / (-7(1-7t))^7
Simplifying the expression inside the integral:
∫ du / (-7(1-7t))^7 = ∫ du / (-7u)^7
Note that (-7u)^7 = -(7u)^7, so we can rewrite the integral as:
∫ du / (-7u)^7 = -∫ du / (7u)^7
Next, we can pull out the constant factor 1/7^7 from the denominator since it does not depend on u:
-∫ du / (7u)^7 = -1/(7^7) ∫ du / u^7
Now, we can integrate with respect to u:
-1/(7^7) ∫ du / u^7 = -1/(7^7) * (-1/6) * u^(-6) + C
Simplifying and substituting u back in terms of t:
-1/(7^7) * (-1/6) * u^(-6) + C = (1/(7^7 * 6)) * (1 - 7t)^(-6) + C
Therefore, the final result is:
∫ dt/(1-7t)^7 = (1/(7^7 * 6)) * (1 - 7t)^(-6) + C, where C is the constant of integration.
To evaluate the integral, we are given the substitution u = 1-7t. To find the new limits of integration and the new differential, we need to express t and dt in terms of u.
First, let's solve the equation u = 1-7t for t:
t = (1-u)/7
Next, we need to find the differential dt in terms of du:
dt = (dt/du) * du
To find the derivative (dt/du), we can differentiate the equation u = 1-7t with respect to t:
1 = -7 * (dt/du)
(dt/du) = -1/7
Now, we can substitute t and dt in the original integral with the new variables u and du:
∫ dt/(1-7)^7 = ∫ (dt/du) * du / (1-7t)^7
Substituting the values we found:
∫ (-1/7) * du / (1-7((1-u)/7))^7 = ∫ (-1/7) * du / (1-u)^7
Now, we can simply evaluate the integral:
∫ (-1/7) * du / (1-u)^7 = (-1/7) * (1/(-6)) * (1-u)^(-6) + C
Simplifying further, we get:
= 1/(42*(1-u)^6) + C
Therefore, the value of the integral ∫ dt/(1-7)^7, with the substitution u=1-7t is 1/(42*(1-u)^6) + C.