# Integral: e^{(ax)} cos(bx) - Solve with Integration by Parts

• The Bob
In summary, the individual is seeking help with the integral of e^(ax) cos(bx). They have already attempted to solve it but are stuck and need assistance. Another individual, Office_Shredder, provides a suggested method of taking the integral on the right-hand side, which leads to a negative result. The individual thanks Office_Shredder for their help.
The Bob
Hi all,

I am having problems with the integral:

$$\int e^{(ax)} cos(bx) dx$$

I have got to $$\frac{e^{ax} sin(bx)}{b} - \int \frac{a e^{ax} sin(bx)}{b} dx$$

After this I can only see myself going around in circles.

Any help would be appreciated.

Cheers,

The Bob said:
$$\int e^{(ax)} cos(bx) dx$$

I have got to $$\frac{e^{ax} sin(bx)}{b} - \int \frac{a e^{ax} sin(bx)}{b} dx$$

The integrals should be:

$$\int e^{(ax)} cos(bx) dx$$

and

$$\frac{e^{ax} sin(bx)}{b} - \int \frac{a e^{ax} sin(bx)}{b} dx$$

LaTex doesn't seem to be editable anymore.

Go again, taking the integral on the RHS Since you get a -cos(x), where you would normally have subtraction from your integration by parts you get addition, except the whole thing is negative anyway, so your new integral (which is a bunch of constants times eaxcos(bx) ) turns out negative. Add that to both sides, and multiply/divide by constants to isolate your original integral

Office_Shredder said:
Go again, taking the integral on the RHS Since you get a -cos(x), where you would normally have subtraction from your integration by parts you get addition, except the whole thing is negative anyway, so your new integral (which is a bunch of constants times eaxcos(bx) ) turns out negative. Add that to both sides, and multiply/divide by constants to isolate your original integral
Cheers Office_Shredder. Sorry for the late reply, I have been very busy recently.

Thanks so much again ,

## 1. What is Integration by Parts?

Integration by Parts is a method used in calculus to find the integral of a product of two functions. It involves using the product rule for differentiation in reverse.

## 2. How do you determine which function to integrate and which to differentiate?

When using Integration by Parts, you can use the acronym "LIATE" to help determine which function to integrate and which to differentiate. "LIATE" stands for Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, and Exponential. The functions that come first in this list should be integrated, while the ones that come last should be differentiated.

## 3. How do you apply Integration by Parts to solve "Integral: e^(ax) cos(bx)"?

To apply Integration by Parts to solve "Integral: e^(ax) cos(bx)", you would first let u = e^(ax) and dv = cos(bx). Then, you would find the derivative of u and the integral of dv. Finally, you would use the formula ∫uv = uv - ∫vdu to solve for the integral.

## 4. What if the integral involves both sine and cosine?

If the integral involves both sine and cosine, you can use the same process as described in question 3, but you will have to apply Integration by Parts twice to fully solve the integral.

## 5. Are there any other methods to solve integrals besides Integration by Parts?

Yes, there are other methods to solve integrals such as substitution, trigonometric substitution, and partial fractions. The method used will depend on the specific integral and the preferences of the person solving it.

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