Using integration by parts, find the following integrals: (a) ∫ x ln x dx (b) ∫ x sin x dx (c) ∫ x² e³ˣ dx. Find the following integrals: (a) ∫ (x+1)/(x²-3x+2) dx (b) ∫ (x²+1)/(x+2)² dx

b) To evaluate $\int x \sin x \, dx$, we use integration by parts.

c) To evaluate $\int x^2 e^{3x} \, dx$, we use integration by parts twice.

Choose $u = x$ and $dv = e^{3x} \, dx$.

Question 2 (b)

a) To evaluate $\int \frac{x+1}{x^2 - 3x + 2} \, dx$, we use partial fraction decomposition.

$x^2 - 3x + 2 = (x-1)(x-2)$.

Multiply both sides by $(x-1)(x-2)$:

Set $x=1$: $1+1 = A(1-2) \implies 2 = -A \implies A = -2$.
Set $x=2$: $2+1 = B(2-1) \implies 3 = B \implies B = 3$.

b) To evaluate $\int \frac{x^2+1}{(x+2)^2} \, dx$, we can manipulate the numerator.

Substitute this into the numerator:

Note that $dx = dy$: