Integrals

∫eˣdx = eˣ + C, ∫(1/x)dx = ln|x| + C, ∫cos(x)dx = sin(x) + C, ∫sec²(x)dx = tan(x) + C

Standard integration formulas: • ∫eˣdx = eˣ + C ✓ • ∫(1/x)dx = ln|x| + C ✓ (note the absolute value) • ∫cos(x)dx = sin(x) + C ✓ • ∫sec²(x)dx = tan(x) + C ✓ • ∫x⁻¹dx should be ln|x| + C, not ln(x) + C

x⁵ - x³ + 7x + C

∫(5x⁴ - 3x² + 7)dx = 5∫x⁴dx - 3∫x²dx + 7∫dx = 5(x⁵/5) - 3(x³/3) + 7x + C = x⁵ - x³ + 7x + C

Integration by parts

For ∫x·ln(x)dx, we have a product of two different types of functions (polynomial and logarithmic). Integration by parts is the appropriate method when dealing with products like this. We would choose u = ln(x) and dv = x dx.

3sin(x) + 2cos(x) + C

∫(3cos(x) - 2sin(x))dx = 3∫cos(x)dx - 2∫sin(x)dx = 3sin(x) - 2(-cos(x)) + C = 3sin(x) + 2cos(x) + C