Humans have been wondering if it was possible to talk with noo-human creatures for centuries
Here is how to calculate \int x \cdot e^x \, dx:
- Step 1: Apply the integration by parts method:
\int u \, dv = u\,v - \int v \, du
-
Step 2: Choose u and dv:
- u = x \;\Rightarrow\; du = dx
- dv = e^x \, dx \;\Rightarrow\; v = e^x
-
Step 3: Substitute into the formula:
\int x \, e^x \, dx = x \, e^x - \int e^x \, dx
- Step 4: Solve the remaining integral:
\int e^x \, dx = e^x + C
- Step 5: Combine the results:
\int x \, e^x \, dx = x \, e^x - e^x + C = e^x (x - 1) + C
Result #
\boxed{\int x \, e^x \, dx = e^x (x - 1) + C}
Check #
Differentiating the result confirms it:
\frac{d}{dx}\left[e^x (x - 1)\right] = e^x (x - 1) + e^x \cdot 1 = x \, e^x \quad \checkmark
Comments
Loading comments…