Integrace per partes: těžší příklady (článek)

Příklad 1

integral, e, start superscript, x, end superscript, sine, x, d, x, equals

(Možnost A)
e, start superscript, x, end superscript, left parenthesis, sine, x, plus, cosine, x, right parenthesis, plus, C
(Možnost B)
start fraction, 1, divided by, 2, end fraction, e, start superscript, x, end superscript, left parenthesis, sine, x, minus, cosine, x, right parenthesis, plus, C
(Možnost C)
2, e, start superscript, x, end superscript, sine, x, plus, C
(Možnost D)
start fraction, e, start superscript, x, end superscript, cosine, x, divided by, 2, end fraction, plus, C

integral, left parenthesis, natural log, x, right parenthesis, squared, d, x, equals

(Možnost A)
x, left parenthesis, natural log, x, right parenthesis, squared, plus, C
(Možnost B)
x, left parenthesis, natural log, x, right parenthesis, squared, minus, 2, natural log, x, plus, C
(Možnost C)
x, left parenthesis, natural log, x, right parenthesis, squared, minus, 2, natural log, x, plus, 2, x, plus, C
(Možnost D)
x, left parenthesis, natural log, x, right parenthesis, squared, minus, 2, x, natural log, x, plus, 2, x, plus, C

integral, x, squared, sine, left parenthesis, pi, x, right parenthesis, d, x, equals

(Možnost A)
start fraction, minus, x, squared, cosine, left parenthesis, pi, x, right parenthesis, plus, x, sine, left parenthesis, pi, x, right parenthesis, divided by, pi, end fraction, plus, start fraction, cosine, left parenthesis, pi, x, right parenthesis, divided by, pi, squared, end fraction, plus, C
(Možnost B)
minus, start fraction, x, squared, cosine, left parenthesis, pi, x, right parenthesis, divided by, pi, end fraction, plus, start fraction, 2, x, sine, left parenthesis, pi, x, right parenthesis, divided by, pi, squared, end fraction, plus, start fraction, 2, cosine, left parenthesis, pi, x, right parenthesis, divided by, pi, cubed, end fraction, plus, C
(Možnost C)
start fraction, minus, x, squared, cosine, left parenthesis, pi, x, right parenthesis, plus, 2, x, sine, left parenthesis, pi, x, right parenthesis, plus, 2, cosine, left parenthesis, pi, x, right parenthesis, divided by, pi, end fraction, plus, C
(Možnost D)
minus, start fraction, x, squared, cosine, left parenthesis, pi, x, right parenthesis, divided by, pi, end fraction, plus, 2, x, sine, left parenthesis, pi, x, right parenthesis, plus, C