What is the integral of
7e^(7t)
Divided By
e^14t+13e^7t+36
Using partial fractions
Substitute t = Log(x)/7
Integral is then proportional to:
Integral of dx/[x^2 + 13 x + 36] =
Integral of dx/[(x+4)(x+9)]
1/[(x+4)(x+9)] = A/(x+4) + B/(x+9)
Multiply both sides by x+4 and take limit x to -4:
1/5 = A
Multiply both sides by x+9 and take limit x to -9:
1/5 = A
-1/5 = B
Thank you so much.
I've reached the last step of
(7/5) ln((x+4)/(x+9)
But how do I substitute the variable 't' back in?
t = Log(x)/7 ---->
x = exp(7 t)
To find the integral of the given expression using partial fractions, follow these steps:
Step 1: Factorize the denominator.
Rewrite the denominator as (e^14t + 13e^7t + 36) = [(e^7t)^2 + 13e^7t + 36]. Notice that this can be factored as (e^7t + 4)(e^7t + 9).
Step 2: Set up the partial fraction decomposition.
The partial fraction decomposition is given by:
7e^(7t)/[(e^7t + 4)(e^7t + 9)] = A/(e^7t + 4) + B/(e^7t + 9)
Step 3: Find the values of A and B.
Multiply both sides of the equation by the denominator to clear the fraction:
7e^(7t) = A(e^7t + 9) + B(e^7t + 4)
Now we can equate the coefficients of like powers of e^7t.
Coefficient of e^7t: 7 = A + B
Constant term: 0 = 9A + 4B
Solve the above system of equations to find the values of A and B.
Step 4: Integrate each term separately.
Now that we have found the values of A and B (by solving the system of equations from Step 3), we can integrate each term separately:
∫[7e^(7t)/[(e^7t + 4)(e^7t + 9)]] dt = A∫[1/(e^7t + 4)] dt + B∫[1/(e^7t + 9)] dt
The integrals of the above terms can be evaluated using integration techniques such as substitution or the natural logarithmic function, depending on the complexity of the expression.
Step 5: Simplify the integral.
After evaluating the integrals from Step 4, you will have the final expression for the integral of the given function.