Integral Properties and Common Integrals

Conditions for using this reference

The definite-integral identities require integrability on every interval involved. The inequalities assume $a\le b$. Reversing limits changes the sign: $\int_b^a f=-\int_a^b f$. An intermediate point $c$ outside $[a,b]$ is allowed only when $f$ is also integrable on the additional intervals.

Antiderivatives apply on intervals within the real domain; never use them to cross a singularity. In the formulas below:

• $a\ne0$ in the exponential and linear-denominator rules; if $a=0$, integrate the resulting constant separately (unless its denominator is zero).
• $ax>0$ for $\ln(ax)$; $x\ne0$ for $\ln|x|$; $ax+b\ne0$ for the rational rule.
• Take $a>0$ in the inverse-trigonometric and inverse-hyperbolic formulas, and $|u|<a$ for the arcsine integrand. The notation $\sin^{-1}$ means arcsine, not reciprocal sine.
• For arbitrary real powers, use $x>0$ and $n\ne-1$; integer powers may extend to negative inputs, excluding zero for negative powers.
• Secant and tangent require $\cos u\ne0$; cosecant and cotangent require $\sin u\ne0$.

Check a lookup by differentiating. For example, $(x\ln(ax)-x)'=\ln(ax)+1-1=\ln(ax)$ on $ax>0$. Linearity does not extend to products: $\int_0^1x^2\,dx=1/3$, whereas $(\int_0^1x\,dx)^2=1/4$.

Linearity of Integration

• Additivity: $\int_{a}^{b} [f(x) + g(x)] \, dx = \int_{a}^{b} f(x) \, dx + \int_{a}^{b} g(x) \, dx$
• Scalar Multiplication: $\int_{a}^{b} cf(x) \, dx = c \int_{a}^{b} f(x) \, dx \quad \text{where } c \text{ is a constant}$

Integrals over Adjacent Intervals

• Combining Intervals: $\int_{a}^{b} f(x) \, dx = \int_{a}^{c} f(x) \, dx + \int_{c}^{b} f(x) \, dx \quad \text{for any value } c$
• Zero Integral: $\int_{a}^{a} f(x) \, dx = 0$

Constants in Integrals

• Constant Function: $\int_{a}^{b} c \, dx = c(b - a) \quad \text{where } c \text{ is a constant}$

Inequality Properties

• Order Preservation: If $f(x) \geq g(x)$ on $[a, b]$: $\int_{a}^{b} f(x) \, dx \geq \int_{a}^{b} g(x) \, dx$
• Non-Negativity: If $f(x) \geq 0$ on $[a, b]$: $\int_{a}^{b} f(x) \, dx \geq 0$
• Bounding the Integral: If $m \leq f(x) \leq M$ on $[a, b]$: $m(b - a) \leq \int_{a}^{b} f(x) \, dx \leq M(b - a)$

Absolute Value Inequality

• Integral Absolute Value: $\left| \int_{a}^{b} f(x) \, dx \right| \leq \int_{a}^{b} |f(x)| \, dx$

Common Integrals

Basic Power and Exponential Functions

• Constant Function: $\int k \, dx = kx + c$
• Power Function (n ≠ -1): $\int x^n \, dx = \frac{1}{n+1}x^{n+1} + c$
• Exponential Function: $\int e^{ax} \, dx = \frac{1}{a}e^{ax} + c$

Logarithmic and Reciprocal Functions

• Reciprocal Function: $\int \frac{1}{x} \, dx = \ln|x| + c$
• Logarithmic Function: $\int \ln(ax) \, dx = x \ln(ax) - x + c$

Trigonometric Functions

• Cosine: $\int \cos(u) \, du = \sin(u) + c$
• Sine: $\int \sin(u) \, du = -\cos(u) + c$
• Secant Squared: $\int \sec^2(u) \, du = \tan(u) + c$

Inverse Trigonometric Functions

• Arcsine (Inverse Sine): $\int \frac{1}{\sqrt{a^2 - u^2}} \, du = \sin^{-1}\left(\frac{u}{a}\right) + c$
• Arctangent (Inverse Tangent): $\int \frac{1}{a^2 + u^2} \, du = \frac{1}{a}\tan^{-1}\left(\frac{u}{a}\right) + c$

Trigonometric Identities Involving Reciprocals

• Secant: $\int \sec(u) \, du = \ln|\sec(u) + \tan(u)| + c$
• Cosecant: $\int \csc(u) \, du = -\ln|\csc(u) + \cot(u)| + c$
• Cosecant Squared: $\int \csc^2(u) \, du = -\cot(u) + c$

Hyperbolic Functions

• Hyperbolic Sine (sinh): $\int \sinh(u) \, du = \cosh(u) + c$
• Hyperbolic Cosine (cosh): $\int \cosh(u) \, du = \sinh(u) + c$

Integration by Substitution

For a function $f(u)$ where $u = g(x)$, the substitution method can be used. For example:

• For $f(u) = \sec(u) \tan(u)$, and $u = g(x)$: $\int \sec(g(x)) \tan(g(x)) g'(x) \, dx = \sec(g(x)) + C$

Integral of a Rational Function

• Simple Rational Function: $\int \frac{1}{ax + b} \, dx = \frac{1}{a} \ln|ax + b| + c$

Integrals Involving Inverse Hyperbolic Functions

• Inverse Hyperbolic Sine: $\int \frac{1}{\sqrt{u^2 + a^2}} \, du = \ln(u + \sqrt{u^2 + a^2}) + c$

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