How would I evaluate the following integral by using integration by parts?
Integral of: (t^3)(e^x)?
You mean (x^3)(e^x)?
x^3 exp(x) dx =
x^3 d[exp(x)] =
d[x^3 exp(x)] - exp(x) d[x^3] =
d[x^3 exp(x)] - 3 x^2 exp(x) dx
So, if you integrate this you get
x^3 exp(x) - 3 Integral of x^2 exp(x) dx
You must repeat this two more times the power on front of the exp(x) will get zero and then you have the answer.
It's simpler to calculate the integral by evaluating the integral of Exp(ax) and then to differentiate the answer three times w.r.t. a in order to get the x^3 in front of the exponential.
To evaluate the integral ∫(t^3)(e^x) using integration by parts, you can follow these steps:
Step 1: Identify u and dv in the equation. In this case, let u = t^3 and dv = e^x dx.
Step 2: Calculate du and v. Differentiate u with respect to t to find du, and integrate dv with respect to x to find v.
→ du = d/dt(t^3) dt = 3t^2 dt
→ v = ∫e^x dx = e^x
Step 3: Apply the integration by parts formula, which states that ∫u dv = uv - ∫v du.
∫(t^3)(e^x) dt = t^3 e^x - ∫e^x (3t^2) dt
Step 4: Simplify the remaining integral and repeat the process.
The new integral is ∫e^x (3t^2) dt. Repeat the integration by parts process by identifying u and dv, finding du and v, and applying the formula.
Let u = 3t^2 and dv = e^x dt
→ du = d/dt(3t^2) dt = 6t dt
→ v = ∫e^x dt = e^x
Using the integration by parts formula again, we have:
∫e^x (3t^2) dt = 3t^2 e^x - ∫e^x (6t) dt
Step 5: Simplify the remaining integral and repeat the process.
The new integral is ∫e^x (6t) dt. Repeat the integration by parts process by identifying u and dv, finding du and v, and applying the formula.
Let u = 6t and dv = e^x dt
→ du = d/dt(6t) dt = 6 dt
→ v = ∫e^x dt = e^x
Using the integration by parts formula one final time, we have:
∫e^x (6t) dt = 6t e^x - ∫e^x (6) dt
Step 6: Simplify the remaining integral.
The integral ∫e^x (6) dt is a constant, so it becomes 6t e^x - 6 ∫e^x dt.
Step 7: Evaluate the integral.
The integral ∫e^x dt is simply e^x.
Therefore, the integral ∫(t^3)(e^x) dt can be evaluated step-by-step as follows:
∫(t^3)(e^x) dt = t^3 e^x - ∫e^x (3t^2) dt
= t^3 e^x - (3t^2 e^x - ∫e^x (6t) dt)
= t^3 e^x - 3t^2 e^x + 6t e^x - 6 ∫e^x dt
= t^3 e^x - 3t^2 e^x + 6t e^x - 6e^x + C
where C represents the constant of integration.
To evaluate the integral of (t^3)(e^x) using integration by parts, we can proceed as follows:
1. Identify u and dv in the given integral:
Let u = t^3
Let dv = e^x dx
2. Determine du and v by differentiating and integrating u and dv, respectively:
Differentiating u, we get du = 3t^2 dt
Integrating dv, we get v = e^x
3. Apply the integration by parts formula: ∫ u dv = uv - ∫ v du
Now, we can substitute the values of u, v, du, and dv into the formula:
∫ (t^3)(e^x) dx = (t^3)(e^x) - ∫ e^x (3t^2 dt)
4. Simplify and obtain the final result:
We can expand the expression (t^3)(e^x) to get t^3e^x and integrate the remaining term ∫ e^x (3t^2 dt) using the power rule for integration. The result will include an arbitrary constant of integration.
So, the solution to the integral of (t^3)(e^x) is:
(t^3)(e^x) - 3∫ t^2(e^x) dt + C, where C represents the constant of integration.