Question

Integración por partes de ∫ x sec^2 (3x) dx

Answer 1

∫u dv = uv - ∫v du

∫x sec²(3x) dx

u = x

du = dx

dv = sec²(3x)dx

∫sec²(3x)dx

z = 3x, dz = 3dx, $dx=\frac{dz}{3}$

∫sec²(z)$\frac{dz}{3}$ = $\frac{1}{3}$ ∫ sec²(z) dz = $\frac{1}{3}$tan(z) = $\frac{1}{3}$tan(3x) + C

v = $\frac{1}{3}$tan(3x)

∫x sec²(3x) dx = $\frac{x}{3}$ tan(3x) - ∫$\frac{1}{3}tan(3x) dx$

-∫$\frac{1}{3}$tan(3x) dx = $-\frac{1}{3}$∫ tan(3x) dx

z = 3x, dz = 3dx, dx = dz/3

$-\frac{1}{3}$∫$\frac{1}{3}$ tan(z) dz = $-\frac{1}{9}$∫ tan(z) dz = $+\frac{1}{9}$ln|cos(3x)| + C

∫x sec²(3x) dx = $\frac{x}{3}$ tan(3x) + $\frac{1}{9}ln|cos(3x)|+C$