Integral Calculus
Integral Calculus tests antiderivatives, the standard integral forms, the three main techniques (substitution, partial fractions, integration by parts), and definite integrals including simple areas.
Standard integrals and techniques
∫xⁿ dx = x^{n+1}/(n+1) + C for n ≠ −1, ∫(1/x) dx = log|x| + C, ∫eˣ dx = eˣ + C, ∫sin x dx = −cos x + C, ∫cos x dx = sin x + C, ∫sec²x dx = tan x + C, and ∫1/(1 + x²) dx = tan⁻¹x + C. Integration by parts is ∫u dv = uv − ∫v du, choosing u in the order inverse trigonometric, logarithmic, algebraic, trigonometric, exponential. For a definite integral, change the variable of integration and change the limits with it; also ∫ₐᵇ f = −∫_b^a f. Area under a curve is obtained by integrating the modulus of the function between the limits.
Worked example
Find ∫x eˣ dx. Take u = x and dv = eˣ dx, so du = dx and v = eˣ. Then ∫x eˣ dx = x eˣ − ∫eˣ dx = x eˣ − eˣ + C = eˣ(x − 1) + C. Differentiating back gives eˣ(x − 1) + eˣ = x eˣ, confirming the answer.
Traps
- Leaving out the constant of integration.
- Keeping the original limits after substituting in a definite integral.
- Losing a sign in the by-parts formula, or choosing u and dv badly.
- Writing ∫(1/x) dx = log x rather than log|x|.