Indefinite Integrals

We've look at The Fundamental Theorem of Calculus Part 1 and The Fundamental Theorem of Calculus Part 2 to calculate definite integrals. There are some cases in which we don't want to calculate an integral on an interval $[a, b]$, but instead, we may be more interested with the general integral itself.

Definition: If $f$ is a function, then the Indefinite Integral of $f$ denoted $\int f(x) \: dx = F(x) + C$ where $F(x) + C$ is any antiderivative of $f$.

We should note that indefinite integration is analogous to that of anti-differentiation, and as a result, we obtain the following list of indefinite integrals:

• If $k$ is a real number, then $\int k f(x) \: dx = k \int f(x) \: dx$.

• If $k$ is a real number, then $\int k \: dx = kx + C$.

• If $n ≠ 1$, then $\int x^n \: dx = \frac{x^{n+1}}{n + 1} + C$.

• $\int \sin x \: dx = -\cos x + C$.

• $\int \cos x \: dx = \sin x + C$.

• $\int \sec ^2 x \: dx = \tan x + C$.

• $\int \sec x \tan x \: dx = \sec x + C$.

• $\int \csc x \cot x \: dx = -\csc x + C$.

• $\int a^x \: dx = \frac{a^x}{\ln a} + C$.

• $\int e^x \: dx = e^x + C$.

• $\int \frac{1}{x} \: dx = \ln \mid x \mid + C$.

• $\int \frac{1}{\sqrt{1 - x^2}} \: dx = \sin ^{-1} x + C$.

• $\int \frac{1}{x^2 + 1} \: dx = \tan ^{-1} x + C$.

Remark: Like when dealing with antiderivatives, it is important to always add a $+C$ for indefinite integrals! Many courses will deduct marks if this detail is forgotten.