Предизвикателство върху интегриране по части (статия)

Задача 1

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

(Избор А)
e, start superscript, x, end superscript, left parenthesis, sine, x, plus, cosine, x, right parenthesis, plus, C
(Избор Б)
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
(Избор В)
2, e, start superscript, x, end superscript, sine, x, plus, C
(Избор Г)
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

(Избор А)
x, left parenthesis, natural log, x, right parenthesis, squared, plus, C
(Избор Б)
x, left parenthesis, natural log, x, right parenthesis, squared, minus, 2, natural log, x, plus, C
(Избор В)
x, left parenthesis, natural log, x, right parenthesis, squared, minus, 2, natural log, x, plus, 2, x, plus, C
(Избор Г)
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

(Избор А)
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
(Избор Б)
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
(Избор В)
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
(Избор Г)
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