Which expression is equal to `int x^5 e^{7x} dx`?
- `1/7 x^5 e^{7x} - 5/7 int x^4 e^{7x} dx`
- `1/7 x^5 e^{7x} - 5/7 int x^5 e^{7x} dx`
- `5/7 x^4 e^{7x} - 5/7 int x^4 e^{7x} dx`
- `5/7 x^4 e^{7x} - 5/7 int x^5 e^{7x} dx`
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Which expression is equal to `int x^5 e^{7x} dx`?
`A`
`u = x^5` | `v′ = e^{7x}` | |
`u′ = 5x^4` | `v = 1/7 e^{7x}` |
`int uv′` | `= uv – int u′ v \ dx` | |
`= 1/7 x^5 e^{7x} – 5/7 int x^4 e^{7x} dx` |
`=>\ A`
Use integration by parts to find `int x e^(3x) dx`. (2 marks)
`(x e^(3x))/3-(1)/(9) e^(3x) + C`
`u` | `= x` | `\ \ \ \ u^{′}` | `= 1` |
`v^{′}` | `= e^(3x)` | `v` | `= (1)/(3) e^(3x)` |
`int x e^(3x) dx` | `= uv-int u^{′} v \ dx` |
`= (xe^(3x))/(3)-(1)/(3) int e ^(3x) dx` | |
`= (x e^(3x))/3-(1)/(9) e^(3x) + C` |
Find `int x e^(-2x)\ dx.` (3 marks)
`−1/2 xe^(−2x) – 1/4 e^(−2x) + c`
`text(Integrating by parts:)`
`text(Let)` | `u` | `= x` | `vprime` | `= e^(−2x)` |
`uprime` | `= 1` | `v` | `= −1/2e^(−2x)` |
`int xe^(−2x)dx` | `= x · −1/2 e^(−2x) + 1/2int e^(−2x)dx` |
`= −1/2 xe^(−2x) – 1/4 e^(−2x) + c` |
Evaluate `int_0^2 te^-t\ dt.` (3 marks)
`1 – 3/e^2`
`u` | `=t` | `u′` | `=1` |
`v` | `=-e^-t\ dt` | `v′` | `=e^-t` |
`int_0^2 te^-t\ dt =` | `[t (-e^-t)]_0^2 – int_0^2 1 xx (-e^-t)\ dt` |
`=` | `[(-2e^-2)-0] – [e^-t]_0^2` |
`=` | `-2/e^2 – (1/e^2 – 1)` |
`=` | `1 – 3/e^2` |
Find `int x e^(2x)\ dx.` (2 marks)
`(x e^(2x))/2 – e^(2x)/4 + c`
`u` | `=x` | `\ \ \ \ u′` | `=1` |
`v′` | `= e^(2x)` | `v` | `= e^(2x)/2` |
`:.int xe^(2x)\ dx` | `=x * e^(2x)/2 – int e^(2x)/2 * 1\ dx` |
`=(xe^(2x))/2 – e^(2x)/4 + c` |