Standard Integrals Flashcards
What is the integral of x^n?
∫ xⁿ dx ⇒ xⁿ⁺¹/(n + 1) + C (n ≠ –1)
What is the integral of 1/x?
∫ 1/x dx ⇒ ln|x| + C
What is the integral of e^x?
∫ eˣ dx ⇒ eˣ + C
What is the integral of a^x?
∫ aˣ dx ⇒ aˣ/ln(a) + C
What is the integral of ln(x)?
∫ ln(x) dx ⇒ x·ln(x) – x + C
What is the integral of sin(x)?
∫ sin(x) dx ⇒ –cos(x) + C
What is the integral of cos(x)?
∫ cos(x) dx ⇒ sin(x) + C
What is the integral of tan(x)?
∫ tan(x) dx ⇒ –ln|cos(x)| + C
What is the integral of cot(x)?
∫ cot(x) dx ⇒ ln|sin(x)| + C
What is the integral of sec(x)?
∫ sec(x) dx ⇒ ln|sec(x)+tan(x)| + C
What is the integral of csc(x)?
∫ csc(x) dx ⇒ –ln|csc(x)+cot(x)| + C
What is the integral of 1/√(1–x²)?
∫ 1/√(1–x²) dx ⇒ arcsin(x) + C
What is the integral of 1/(1+x²)?
∫ 1/(1+x²) dx ⇒ arctan(x) + C
What is the integral of 1/[|x|√(x²–1)]?
∫ 1/[|x|√(x²–1)] dx ⇒ arcsec(x) + C
What is the integral of –1/√(1–x²)?
∫ –1/√(1–x²) dx ⇒ arccos(x) + C
What is the integral of –1/(1+x²)?
∫ –1/(1+x²) dx ⇒ arccot(x) + C
What is the integral of –1/[|x|√(x²–1)]?
∫ –1/[|x|√(x²–1)] dx ⇒ arccsc(x) + C
What is the integral of the sum of two functions f(x) and g(x)?
∫ [f(x)+g(x)] dx ⇒ ∫f(x) dx + ∫g(x) dx
What is the integral of a constant multiplied by a function?
∫ c·f(x) dx ⇒ c·∫f(x) dx
What is the integral of f(g(x))·g′(x)?
∫ f(g(x))·g′(x) dx ⇒ ∫f(u) du (u = g(x))
What is the formula for integration by parts?
∫ u·dv ⇒ u·v – ∫v·du
What is the parametric integral of y dx?
∫ y dx ⇒ ∫ y(t)·x′(t) dt + C
What is the parametric integral of x dy?
∫ x dy ⇒ ∫ x(t)·y′(t) dt + C
