Calculate The Integral E 5x Cos 7x

This calculator computes the integral of e^(5x) * cos(7x) using integration by parts. The result is expressed in terms of elementary functions and includes a visualization of the integrand and integral function.

How to Calculate the Integral

The integral of e^(5x) * cos(7x) is a common calculus problem that requires integration by parts. The general approach involves:

Identifying the integrand as a product of two functions: u(x) = e^(5x) and v(x) = cos(7x)
Applying integration by parts formula: ∫u dv = uv - ∫v du
Choosing u and dv appropriately to simplify the integral
Repeating the process until the integral can be evaluated

The integral is calculated using the formula:

∫e^(5x)cos(7x) dx = (e^(5x)(5cos(7x) + 7sin(7x)))/(25 + 49) + C

Where C is the constant of integration.

Step-by-Step Solution

To compute ∫e^(5x)cos(7x) dx, we use integration by parts twice:

First integration by parts:
- Let u = e^(5x) ⇒ du = 5e^(5x) dx
- Let dv = cos(7x) dx ⇒ v = (1/7)sin(7x)
- ∫e^(5x)cos(7x) dx = (e^(5x)/7)sin(7x) - ∫(5e^(5x)/7)sin(7x) dx
Second integration by parts on the remaining integral:
- Let u = sin(7x) ⇒ du = 7cos(7x) dx
- Let dv = (5e^(5x)/7) dx ⇒ v = (5e^(5x)/49)
- ∫(5e^(5x)/7)sin(7x) dx = (5e^(5x)/49)sin(7x) - ∫(5e^(5x)/49)(7cos(7x)) dx
Combine the results and simplify:
- ∫e^(5x)cos(7x) dx = (e^(5x)/7)sin(7x) - (5e^(5x)/49)sin(7x) + (5e^(5x)/49)cos(7x) + C
- Factor out e^(5x): ∫e^(5x)cos(7x) dx = e^(5x)[(1/7 - 5/49)sin(7x) + (5/49)cos(7x)] + C
- Simplify coefficients: ∫e^(5x)cos(7x) dx = e^(5x)[(2/49)sin(7x) + (5/49)cos(7x)] + C

Worked Example

Let's compute the definite integral from 0 to π/2:

∫[0 to π/2] e^(5x)cos(7x) dx = e^(5x)[(2/49)sin(7x) + (5/49)cos(7x)] evaluated from 0 to π/2

At x = π/2:

e^(5π/2)[(2/49)sin(7π/2) + (5/49)cos(7π/2)] = e^(5π/2)[(2/49)(1) + (5/49)(0)] = (2/49)e^(5π/2)

At x = 0:

e^(0)[(2/49)sin(0) + (5/49)cos(0)] = (5/49)

Final result: (2/49)e^(5π/2) - (5/49) ≈ 0.0408e^(8.6394) - 0.1020

FAQ

What is the integral of e^(5x)cos(7x)?: The integral is e^(5x)[(2/49)sin(7x) + (5/49)cos(7x)] + C, where C is the constant of integration.
How do I compute this integral?: Use integration by parts twice, choosing u = e^(5x) and dv = cos(7x) for the first integration, then u = sin(7x) and dv = (5e^(5x)/7) for the second integration.
Can this integral be expressed in terms of simpler functions?: Yes, the result is expressed in terms of elementary functions involving e^(5x), sin(7x), and cos(7x).
What are the assumptions in this calculation?: The calculation assumes x is a real number and that the integral converges. The result is valid for all real x.