How do you integrate #f(x)=3^(5x-1)/e^(4x)# using the quotient rule?

Answer 1

#int 3^(5x-1)/e^(4x) dx = 3^(5x-1)/(e^(4x)(5ln3-4))+C#

Note that:

#3^(5x-1)/e^(4x) = (e^(ln3))^(5x-1)*e^(-4x) = e^((5ln3 -4)x)*e^(-ln3) = 1/3 e^((5ln3 -4)x#

So:

#int 3^(5x-1)/e^(4x) dx = 1/3 int e^((5ln3 -4)x)dx = e^((5ln3 -4)x)/(3(5ln3-4))+C = 3^(5x-1)/(e^(4x)(5ln3-4))+C#
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To integrate ( f(x) = \frac{{3^{5x-1}}}{{e^{4x}}} ) using the quotient rule, first, rewrite the function as:

[ f(x) = 3^{5x-1} \cdot e^{-4x} ]

Now, integrate using the quotient rule for integration, which states that for functions ( u(x) ) and ( v(x) ), the integral of ( \frac{{u(x)}}{{v(x)}} ) can be found by:

[ \int \frac{{u(x)}}{{v(x)}} , dx = \frac{{\int u(x) , dx}}{{\int v(x) , dx}} ]

Apply this rule to ( f(x) ):

[ \int \frac{{3^{5x-1}}}{{e^{4x}}} , dx = \frac{{\int 3^{5x-1} , dx}}{{\int e^{-4x} , dx}} ]

Now, integrate each part:

[ \int 3^{5x-1} , dx = \frac{1}{5} \cdot \frac{3^{5x}}{5\ln(3)} + C_1 ]

[ \int e^{-4x} , dx = -\frac{1}{4} \cdot e^{-4x} + C_2 ]

Where ( C_1 ) and ( C_2 ) are constants of integration.

Now, combine these results:

[ \int \frac{{3^{5x-1}}}{{e^{4x}}} , dx = \frac{1}{5\ln(3)} \cdot \frac{3^{5x}}{5} \cdot (-\frac{1}{4}) \cdot e^{-4x} + C ]

[ = -\frac{1}{20\ln(3)} \cdot 3^{5x} \cdot e^{-4x} + C ]

Where ( C ) is the constant of integration.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7