NDA · Chapter

Definite Integration

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find any antiderivative and subtract its values at the limits

with

h = \dfrac{b - a}{n}

:

the total area of thin rectangles

Start

F'(x) = f(x)

\int_a^b f(x)\,dx = F(b) - F(a)

Substitution in a definite integralWhen you put

u = g(x)

, change the limits too:

x = a

becomes

u = g(a)

,

x = b

becomes

u = g(b)

; then never go back to

x

Locked

Parts and partial fractions with limits

\int_a^b u\,v\,dx = \left[u\int v\,dx\right]_a^b - \int_a^b \left(u'\int v\,dx\right)dx

; split rational functions first, then evaluateLocked

Basic properties and splitting the interval $\int_a^b = -\int_b^a$ , $\int_a^b = \int_a^c + \int_c^b$ ; split at the point where $|x - c|$ or a piecewise function changes formLocked
The property f(a + b − x) $\int_a^b f(x)\,dx = \int_a^b f(a + b - x)\,dx$ ;
Odd and even functions, and the 2a property $\int_{-a}^{a} f(x)\,dx$ is $0$ for odd $f$

Differentiating an integral $\frac{d}{dx}\int_a^{g(x)} f(t)\,dt = f(g(x))\,g'(x)$ : the integral of a derivative and the derivative of an integral undo each otherLocked
Periodic integrandsIf $f$ has period $T$ , every stretch of length $T$ gives the same integral, so $\int_0^{nT} f = n\int_0^T f$
Integrands with [x] and |x|Split the interval where $[x]$ jumps or the expression inside $|\ |$ changes sign; on each piece the integrand is a plain functionLocked

add the two forms when they sum to something simple, as in

\int_0^{\pi/2}\dfrac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}}\,dx = \dfrac{\pi}{4}

Locked

and

2\int_0^a f(x)\,dx

for even

f

;

\int_0^{2a} f(x)\,dx = 2\int_0^a f(x)\,dx

when

f(2a - x) = f(x)

Locked