Задачник Кузнецова Интегралы Задачи 5-8

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Содержание

1 Задачи 5
2 Задачи 6
3 Задачи 7
4 Задачи 8

Задачи 5

Найти неопределенные интегралы.

Задача	Условие	Задача	Условие
5-1	$\int \frac{x^3+1}{x^2-x} dx$	5-2	$\int \frac{3x^3+1}{x^2-1} dx$
5-3	$\int \frac{x^3-17}{x^2-4x+3} dx$	5-4	$\int \frac{2x^3+5}{x^2-x-2} dx$
5-5	$\int \frac{2x^3-1}{x^2+x-6} dx$	5-6	$\int \frac{3x^3+25}{x^2+3x+2} dx$
5-7	$\int \frac{x^3+2x^2+3}{(x-1)(x-2)(x-3)} dx$	5-8	$\int \frac{3x^3+2x^2+1}{(x+2)(x-2)(x-1)} dx$
5-9	$\int \frac{x^3}{(x-1)(x+1)(x+2)} dx$	5-10	$\int \frac{x^3-3x^2-12}{(x-4)(x-3)(x-2)} dx$
5-11	$\int \frac{x^3-3x^2-12}{(x-4)(x-3)x} dx$	5-12	$\int \frac{4x^3+x^2+2}{x(x-1)(x-2)} dx$
5-13	$\int \frac{3x^3-2}{x^3-x} dx$	5-14	$\int \frac{x^3-3x^2-12}{(x-4)(x-2)x} dx$
5-15	$\int \frac{x^5-x^3+1}{x^2-x} dx$	5-16	$\int \frac{x^5+3x^3-1}{x^2+x} dx$
5-17	$\int \frac{2x^5-8x^3+3}{x^2-2x} dx$	5-18	$\int \frac{3x^5-12x^3-7}{x^2+2x} dx$
5-19	$\int \frac{-x^5+9x^3+4}{x^2+3x} dx$	5-20	$\int \frac{-x^5+25x^3+1}{x^2+5x} dx$
5-21	$\int \frac{x^3-5x^2+5x+23}{(x-1)(x+1)(x-5)} dx$	5-22	$\int \frac{x^5+2x^4-2x^3+5x^2-7x+9}{(x+3)(x-1)x} dx$
5-23	$\int \frac{2x^4-5x^2-8x-8}{x(x-2)(x+2)} dx$	5-24	$\int \frac{4x^4+2x^2-x-3}{x(x-1)(x+1)} dx$
5-25	$\int \frac{3x^4+3x^3-5x^2+2}{x(x-1)(x+2)} dx$	5-26	$\int \frac{2x^4+2x^3-41x^2+20}{x(x-4)(x+5)} dx$
5-27	$\int \frac{x^5-x^4-6x^3+13x+6}{x(x-3)(x+2)} dx$	5-28	$\int \frac{3x^3-x^2-12x-2}{x(x+1)(x-2)} dx$
5-29	$\int \frac{2x^4+2x^3-3x^2+2x-9}{x(x-1)(x+3)} dx$	5-30	$\int \frac{2x^3-x^2-7x-12}{x(x-3)(x+1)} dx$
5-31	$\int \frac{2x^3-40x-8}{x(x+4)(x-2)} dx$

Задачи 6

Вычислить неопределенные интегралы.

Задача	Условие	Задача	Условие
6-1	$\int \frac{x^3+6x^2+13x+9}{(x+1)(x+2)^3}dx$	6-2	$\int \frac{x^3+6x^2+13x+8}{x(x+2)^3}dx$
6-3	$\int \frac{x^3-6x^2+13x-6}{(x+2)(x-2)^3}dx$	6-4	$\int \frac{x^3+6x^2+14x+10}{(x+1)(x+2)^3}dx$
6-5	$\int \frac{x^3-6x^2+11x-10}{(x+2)(x-2)^3}dx$	6-6	$\int \frac{x^3+6x^2+11x+7}{(x+1)(x+2)^3}dx$
6-7	$\int \frac{2x^3+6x^2+7x+1}{(x-1)(x+1)^3}dx$	6-8	$\int \frac{x^3+6x^2+10x+10}{(x-1)(x+2)^3}dx$
6-9	$\int \frac{2x^3+6x^2+7x+2}{x(x+1)^3}dx$	6-10	$\int \frac{x^3-6x^2+13x-8}{x(x-2)^3}dx$
6-11	$\int \frac{x^3-6x^2+13x-7}{(x+1)(x-2)^3}dx$	6-12	$\int \frac{x^3-6x^2+14x-6}{(x+1)(x-2)^3}dx$
6-13	$\int \frac{x^3-6x^2+10x-10}{(x+1)(x-2)^3}dx$	6-14	$\int \frac{x^3+x+2}{(x+2)x^3}dx$
6-15	$\int \frac{3x^3+9x^2+10x+2}{(x-1)(x+1)^3}dx$	6-16	$\int \frac{2x^3+x+1}{(x+1)x^3}dx$
6-17	$\int \frac{2x^3+6x^2+7x+4}{(x+2)(x+1)^3} dx$	6-18	$\int \frac{2x^3+6x^2+5x}{(x+2)(x+1)^3}dx$
6-19	$\int \frac{2x^3+6x^2+7x}{(x-2)(x+1)^3}dx$	6-20	$\int \frac{2x^3+6x^2+5x+4}{(x-2)(x+1)^3}dx$
6-21	$\int \frac{x^3+6x^2+4x+24}{(x-2)(x+2)^3}dx$	6-22	$\int \frac{x^3+6x^2+14x+4}{(x-2)(x+2)^3}dx$
6-23	$\int \frac{x^3+6x^2+18x-4}{(x-2)(x+2)^3}dx$	6-24	$\int \frac{x^3+6x^2+10x+12}{(x-2)(x+2)^3}dx$
6-25	$\int \frac{x^3-6x^2+14x-4}{(x-2)(x+2)^3}dx$	6-25(2)	$\int \frac{x^3-6x^2+14x-4}{(x+2)(x-2)^3}dx$
6-26	$\int \frac{x^3+6x^2+15x+2}{(x-2)(x+2)^3}dx$	6-27	$\int \frac{2x^3-6x^2+7x-4}{(x-2)(x-1)^3}dx$
6-28	$\int \frac{2x^3-6x^2+7x}{(x+2)(x-1)^3}dx$	6-29	$\int \frac{x^3+6x^2-10x+52}{(x-2)(x+2)^3}dx$
6-30	$\int \frac{x^3-6x^2+13x-6}{(x-2)(x+2)^3}dx$	6-31	$\int \frac{x^3+6x^2+13x+6}{(x-2)(x+2)^3}dx$

Задачи 7

Найти неопределенные интегралы.

Задача	Условие	Задача	Условие
7-1	$\int \frac{x^3+4x^2+4x+2}{(x+1)^2\left(x^2+x+1\right)}dx$	7-2	$\int \frac{x^3+4x^2+3x+2}{(x+1)^2\left(x^2+1\right)} dx$
7-3	$\int \frac{2x^3+7x^2+7x-1}{(x+2)^2\left(x^2+x+1\right)} dx$	7-4	$\int \frac{2x^3+4x^2+2x-1}{(x+1)^2\left(x^2+2x+2\right)} dx$
7-5	$\int \frac{x^3+6x^2+9x+6}{(x+1)^2\left(x^2+2x+2\right)} dx$	7-6	$\int \frac{2x^3+11x^2+16x+10}{(x+2)^2\left(x^2+2x+3\right)} dx$
7-7	$\int \frac{3x^3+6x^2+5x-1}{(x+1)^2\left(x^2+2\right)} dx$	7-8	$\int \frac{x^3+9x^2+21x+21}{(x+3)^2\left(x^2+3\right)} dx$
7-9	$\int \frac{x^3+6x^2+8x+8}{(x+2)^2\left(x^2+4\right)} dx$	7-10	$\int \frac{x^3+5x^2+12x+4}{(x+2)^2\left(x^2+4\right)} dx$
7-11	$\int \frac{2x^3-4x^2-16x-12}{(x-1)^2\left(x^2+4x+5\right)} dx$	7-12	$\int \frac{-3x^3+13x^2-13x+1}{(x-2)^2\left(x^2-x+1 \right)} dx$
7-13	$\int \frac{x^3+2x^2+10x}{(x+1)^2\left(x^2-x+1\right)} dx$	7-14	$\int \frac{3x^3+x+46}{(x-1)^2\left(x^2+9 \right)} dx$
7-15	$\int \frac{4x^3+24x^2+20x-28}{(x+3)^2\left(x^2+2x+2\right)} dx$	7-16	$\int \frac{2x^3+3x^2+3x+2}{\left(x^2+x+1\right)\left(x^2+1\right)} dx$
7-17	$\int \frac{x^3+x+1}{\left(x^2+x+1\right)\left(x^2+1\right)} dx$	7-18	$\int \frac{x^2+x+3}{\left(x^2+x+1\right)\left(x^2+1\right)} dx$
7-19	$\int \frac{2x^3+4x^2+2x+2}{\left(x^2+x+1\right)\left(x^2+x+2\right)} dx$	7-20	$\int \frac{2x^3+7x^2+7x+9}{\left(x^2+x+1\right)\left(x^2+x+2\right)} dx$
7-21	$\int \frac{4x^2+3x+4}{\left(x^2+1\right)\left(x^2+x+1\right)} dx$	7-22	$\int \frac{3x^3+4x^2+6x}{\left(x^2+2 \right)\left(x^2+2x+2 \right)} dx$
7-23	$\int \frac{2x^2-x+1}{\left(x^2-x+1 \right)\left(x^2+1 \right)} dx$	7-24	$\int \frac{x^3+x^2+1}{\left(x^2-x+1 \right)\left(x^2+1 \right)} dx$
7-25	$\int \frac{x^3+x+1}{\left(x^2-x+1 \right)\left(x^2+1 \right)} dx$	7-26	$\int \frac{2x^3+2x+1}{\left(x^2-x+1 \right)\left(x^2+1 \right)} dx$
7-27	$\int \frac{x^3+2x^2+x+1}{\left(x^2+x+1 \right)\left(x^2+1 \right)} dx$	7-28	$\int \frac{x+4}{\left(x^2+x+2 \right)\left(x^2+2 \right)} dx$
7-29	$\int \frac{2x^3+2x^2+2x+1}{\left(x^2+x+1 \right)\left(x^2+1 \right)} dx$	7-30	$\int \frac{3x^3+7x^2+12x+6}{\left(x^2+x+3 \right)\left(x^2+2x+3 \right)} dx$
7-31	$\int \frac{2x^3+3x^2+3x+2}{\left(x^2+x+1 \right)\left(x^2+1 \right)} dx$

Задачи 8

Вычислить определенные интегралы.

Задача	Условие	Задача	Условие
8-1	$\int\limits_{\frac{\pi}{2}}^{2\operatorname{arctg}{2}} \frac{dx}{\sin^{2}{x}\left(1-\cos{x}\right)}$	8-2	$\int\limits_0^{\frac{\pi}{2}} \frac{\cos{x}dx}{2+\cos{x}}$
8-3	$\int\limits_{\frac{\pi}{2}}^{2\operatorname{arctg}{2}} \frac{dx}{\sin^{2}{x}\left(1+\cos{x}\right)}$	8-4	$\int\limits_{2\operatorname{arctg}{\frac{1}{2}}}^{\frac{\pi}{2}} \frac{\cos{x}dx}{\left(1-\cos{x} \right)^3}$
8-5	$\int\limits_0^{\frac{\pi}{2}} \frac{\cos{x}-\sin{x}}{\left(1+\sin{x}\right)^2}dx$	8-6	$\int\limits_{2\operatorname{arctg}{2}}^{2\operatorname{arctg}{3}} \frac{dx}{\cos{x}\left(1-\cos{x}\right)}$
8-7	$\int\limits_{2\operatorname{arctg}{\frac{1}{3}}}^{2\operatorname{arctg}{\frac{1}{2}}} \frac{dx}{\sin{x}\left(1-\sin{x} \right)}$	8-8	$\int\limits_{2\operatorname{arctg}{\frac{1}{2}}}^{\frac{\pi}{2}} \frac{dx}{\left(1+\sin{x}-\cos{x} \right)^2}$
8-9	$\int\limits_0^{\frac{\pi}{2}} \frac{\cos{x}dx}{5+4\cos{x}}$	8-10	$\int\limits_0^{\frac{2\pi}{3}} \frac{1+\sin{x}}{1+\cos{x}+\sin{x}}dx$
8-11	$\int\limits_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{\cos{x}dx}{1+\sin{x}-\cos{x}}$	8-12	$\int\limits_0^{\frac{\pi}{2}} \frac{\left(1+\cos{x} \right)dx}{1+\sin{x}+\cos{x}}$
8-13	$\int\limits_0^{\frac{\pi}{2}} \frac{\sin{x} dx}{1+\sin{x}+\cos{x}}$	8-14	$\int\limits_0^{2\operatorname{arctg}{\frac{1}{2}}} \frac{1+\sin{x}}{\left(1-\sin{x}\right)^2}dx$
8-15	$\int\limits_0^{\frac{\pi}{2}} \frac{\cos{x}dx}{1+\sin{x}+\cos{x}}$	8-16	$\int\limits_0^{2\operatorname{arctg}{\frac{1}{3}}} \frac{\cos{x}dx}{\left(1-\sin{x}\right)\left(1+\cos{x} \right)}$
8-17	$\int\limits_{-\frac{2\pi}{3}}^0 \frac{\cos{x}dx}{1+\cos{x}-\sin{x}}$	8-18	$\int\limits_{-\frac{\pi}{2}}^0 \frac{\cos{x}dx}{\left(1+\cos{x}-\sin{x}\right)^2}$
8-19	$\int\limits_0^{\frac{\pi}{2}} \frac{\cos{x}dx}{\left(1+\cos{x}+\sin{x} \right)^2}$	8-20	$\int\limits_0^{2\operatorname{arctg}{\frac{1}{2}}} \frac{\left(1-\sin{x}\right)dx}{\cos{x}\left(1+\cos{x}\right)}$
8-21	$\int\limits_0^{\frac{\pi}{2}} \frac{\sin{x}dx}{\left(1+\sin{x}\right)^2}$	8-22	$\int\limits_0^{\frac{\pi}{2}} \frac{\sin{x}dx}{\left(1+\cos{x}+\sin{x}\right)^2}$
8-23	$\int\limits_{-\frac{\pi}{2}}^0 \frac{\sin{x}dx}{\left(1+\cos{x}-\sin{x}\right)^2}$	8-24	$\int\limits_{-\frac{2\pi}{3}}^0 \frac{\cos^{2}{x}dx}{\left(1+\cos{x}-\sin{x}\right)^2}$
8-25	$\int\limits_0^{\frac{\pi}{2}} \frac{\sin^{2}{x}dx}{\left(1+\cos{x}+\sin{x}\right)^2}$	8-26	$\int\limits_0^{\frac{2\pi}{3}} \frac{\cos^{2}{x}dx}{\left(1+\cos{x}+\sin{x}\right)^2}$
8-27	$\int\limits_{\frac{\pi}{2}}^{2\operatorname{arctg}{2}} \frac{dx}{\sin{x}\left(1+\sin{x}\right)}$	8-28	$\int\limits_0^{\frac{\pi}{2}} \frac{dx}{\left(1+\cos{x}+\sin{x}\right)^2}$
8-29	$\int\limits_0^{\frac{\pi}{2}} \frac{\sin{x}dx}{2+\sin{x}}$	8-30	$\int\limits_0^{\frac{\pi}{4}} \frac{dx}{\cos{x}\left(1+\cos{x} \right)}$
8-31	$\int\limits_0^{\frac{\pi}{2}} \frac{\sin{x}dx}{5+3\sin{x}}$