Math Problems Vault

Problem 1866

Using the method of undetermined coefficients, find the following integral: $\int \dfrac {2x+3}{(x-2)(x+5)} dx$ .

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Problem 1867

Using the method of undetermined coefficients, find the following integral: $\int \dfrac {x\, dx}{(x+1)(x+2)(x+3)}$ .

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Problem 1868

Using the method of undetermined coefficients, find the following integral: $\int \dfrac {x^{10}}{x^{2}+x-2} dx$ .

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Problem 1869

Using the method of undetermined coefficients, find the following integral: $\int \dfrac {x^{3}+1}{x^{3}-5x^{2}+6x} dx$ .

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Problem 1870

Using the method of undetermined coefficients, find the following integral: $\int \dfrac {x^{4}}{x^{4}+5x^{2}+4} dx$ .

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Problem 1871

Using the method of undetermined coefficients, find the following integral: $\int \dfrac {x\, dx}{x^{3}-3x+2}$ .

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Problem 1872

Using the method of undetermined coefficients, find the following integral: $\int \dfrac {x^{2}+1}{(x+1)^{2}(x-1)} dx$ .

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Problem 1873

Using the method of undetermined coefficients, find the following integral: $\int \left(\dfrac {x}{x^{2}-3x+2}\right)^{2} dx$ .

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Problem 1874

Using the method of undetermined coefficients, find the following integral: $\int \dfrac {dx}{(x+1)(x+2)^{2}(x+3)^{3}}$ .

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Problem 1875

Using the method of undetermined coefficients, find the following integral: $\int \dfrac {dx}{x^{5}+x^{4}-2x^{3}-2x^{2}+x+1}$ .

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Problem 1876

Using the method of undetermined coefficients, find the following integral: $\int \dfrac {x^{2}+5x+4}{x^{4}+5x^{2}+4} dx$ .

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Problem 1877

Using the method of undetermined coefficients, find the following integral: $\int \dfrac {dx}{(x+1)(x^{2}+1)}$ .

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Problem 1878

Using the method of undetermined coefficients, find the following integral: $\int \dfrac {dx}{(x^{2}-4x+4)(x^{2}-4x+5)}$ .

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Problem 1879

Using the method of undetermined coefficients, find the following integral: $\int \dfrac {x\, dx}{(x-1)^{2}(x^{2}+2x+2)}$ .

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Problem 1880

Using the method of undetermined coefficients, find the following integral: $\int \dfrac {dx}{x(1+x)(1+x+x^{2})}$ .

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Problem 1881

Using the method of undetermined coefficients, find the following integral: $\int \dfrac {dx}{x^{3}+1}$ .

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Problem 1882

Using the method of undetermined coefficients, find the following integral: $\int \dfrac {x\, dx}{x^{3}-1}$ .

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Problem 1883

Using the method of undetermined coefficients, find the following integral: $\int \dfrac {dx}{x^{4}-1}$ .

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Problem 1884

Using the method of undetermined coefficients, find the following integral: $\int \dfrac {dx}{x^{4}+1}$ .

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Problem 1885

Using the method of undetermined coefficients, find the following integral: $\int \dfrac {dx}{x^{4}+x^{2}+1}$ .

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Problem 1886

Using the method of undetermined coefficients, find the following integral: $\int \dfrac {dx}{x^{6}+1}$ .

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Problem 1887

Using the method of undetermined coefficients, find the following integral: $\int \dfrac {dx}{(1+x)(1+x^{2})(1+x^{3})}$ .

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Problem 1888

Using the method of undetermined coefficients, find the following integral: $\int \dfrac {dx}{x^{5}-x^{4}+x^{3}-x^{2}+x-1}$ .

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Problem 1889

Using the method of undetermined coefficients, find the following integral: $\int \dfrac {x^{2}\, dx}{x^{4}+3x^{3}+\dfrac {9}{2}x^{2}+3x+1}$ .

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Problem 1890

Under what condition is the integral $\displaystyle \int \dfrac {ax^{2}+bx+c}{x^{3}(x-1)^{2}} dx$ a rational function?

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Problem 1891

Using the Ostrogradsky method, compute the integral: $\int \dfrac {x\, dx}{(x-1)^{2}(x+1)^{3}}$ .

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Remark.

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The so-called Ostrogradsky method refers to the following: for the integral of a proper rational fraction $\dfrac {P(x)}{Q(x)}$ , algebraic means can be used to split it into the sum of a proper fraction and the integral of another proper fraction, in such a way that in the new integrand's proper fraction the degree of the denominator is brought down to its lowest possible state, that is, in the formula

\int \dfrac {P(x)}{Q(x)} dx=\dfrac {P_{1}(x)}{Q_{1}(x)}+\int \dfrac {P_{2}(x)}{Q_{2}(x)} dx \tag {1}

where, if $P(x)$ , $Q(x)$ are known, and the denominator $Q(x)$ is supposed to factor into real linear and quadratic-type factors:

Q(x)=(x-a)^{k}\cdots (x^{2}+px+q)^{m}\cdots ,

where $k,\ldots ,m,\ldots$ are natural numbers. On the right-hand side of formula (1) the denominators are known, of the form:

\begin{align} Q_{1}(x) & =(x-a)^{k-1}\cdots (x^{2}+px+q)^{m-1}\cdots , \\ Q_{2}(x) & =(x-a)\cdots (x^{2}+px+q)\cdots , \end{align}

and satisfy $Q_{1}(x)\cdot Q_{2}(x)=Q(x)$ . Meanwhile $P_{1}(x)$ and $P_{2}(x)$ are polynomials of lower degree than $Q_{1}(x)$ and $Q_{2}(x)$ respectively, and can generally be found by the method of undetermined coefficients. This method of using formula (1) to find the integral is precisely the so-called Ostrogradsky method. For details, see G.M. Fikhtengolts (Peking University translation), A Course of Differential and Integral Calculus, Volume 2, Part 1, p.264. --- Note of the solutions editor.

Problem 1892

Using the Ostrogradsky method, compute the integral: $\int \dfrac {dx}{(x^{3}+1)^{2}}$ .

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Problem 1893

Using the Ostrogradsky method, compute the integral: $\int \dfrac {dx}{(x^{2}+1)^{3}}$ .

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Problem 1894

Using the Ostrogradsky method, compute the integral: $\int \dfrac {x^{2}\, dx}{(x^{2}+2x+2)^{2}}$ .

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Problem 1895

Using the Ostrogradsky method, compute the integral: $\int \dfrac {dx}{(x^{4}+1)^{2}}$ .

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Problem 1896

Using the Ostrogradsky method, compute the integral: $\displaystyle \int \dfrac {x^{2}+3x-2}{(x-1)(x^{2}+x+1)^{2}} dx$ .

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Problem 1897

Using the Ostrogradsky method, compute the integral: $\int \dfrac {dx}{(x^{4}-1)^{3}}$ .

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Problem 1898

Extract the algebraic part of the following integral: $\displaystyle \int \dfrac {x^{2}+1}{(x^{4}+x^{2}+1)^{2}} dx$ .

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Problem 1899

Extract the algebraic part of the following integral: $\displaystyle \int \dfrac {dx}{(x^{3}+x+1)^{3}}$ .

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Problem 1900

Extract the algebraic part of the following integral: $\displaystyle \int \dfrac {4x^{5}-1}{(x^{5}+x+1)^{2}} dx$ .

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Problem 1901

Compute the integral: $\displaystyle \int \dfrac {dx}{x^{4}+2x^{3}+3x^{2}+2x+1}$ .

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Problem 1902

Under what condition is the integral $\displaystyle \int \dfrac {\alpha x^{2}+2\beta x+\gamma }{(ax^{2}+2bx+c)^{2}} dx$ a rational function?

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Problem 1903

Using various methods, calculate the following integral: $\displaystyle \int \dfrac {x^{3}}{(x-1)^{100}} dx$ .

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Problem 1904

Using various methods, calculate the following integral: $\displaystyle \int \dfrac {x\, dx}{x^{8}-1}$ .

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Problem 1905

Using various methods, calculate the following integral: $\displaystyle \int \dfrac {x^{3}\, dx}{x^{8}+3}$ .

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Problem 1906

Using various methods, calculate the following integral: $\displaystyle \int \dfrac {x^{2}+x}{x^{6}+1} dx$ .

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Problem 1907

Using various methods, calculate the following integral: $\displaystyle \int \dfrac {x^{4}-3}{x(x^{8}+3x^{4}+2)} dx$ .

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Problem 1908

Using various methods, calculate the following integral: $\displaystyle \int \dfrac {x^{4}\, dx}{(x^{10}-10)^{2}}$ .

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Problem 1909

Using various methods, calculate the following integral: $\displaystyle \int \dfrac {x^{11}\, dx}{x^{8}+3x^{4}+2}$ .

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Problem 1910

Using various methods, calculate the following integral: $\displaystyle \int \dfrac {x^{9}\, dx}{(x^{10}+2x^{5}+2)^{2}}$ .

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Problem 1911

Using various methods, calculate the following integral: $\displaystyle \int \dfrac {x^{2n-1}}{x^{n}+1} dx$ .

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Problem 1912

Using various methods, calculate the following integral: $\displaystyle \int \dfrac {x^{3n-1}}{(x^{2n}+1)^{2}} dx$ .

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Problem 1913

Using various methods, calculate the following integral: $\displaystyle \int \dfrac {dx}{x(x^{10}+2)}$ .

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Problem 1914

Using various methods, calculate the following integral: $\displaystyle \int \dfrac {dx}{x(x^{10}+1)^{2}}$ .

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Problem 1915

Using various methods, calculate the following integral: $\displaystyle \int \dfrac {1-x^{7}}{x(1+x^{7})} dx$ .

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Problem 1916

Using various methods, calculate the following integral: $\displaystyle \int \dfrac {x^{4}-1}{x(x^{4}-5)(x^{5}-5x+1)} dx$ .

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Problem 1917

Using various methods, calculate the following integral: $\displaystyle \int \dfrac {x^{2}+1}{x^{4}+x^{2}+1} dx$ .

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Problem 1918

Using various methods, calculate the following integral: $\displaystyle \int \dfrac {x^{2}-1}{x^{4}+x^{3}+x^{2}+x+1} dx$ .

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Problem 1919

Using various methods, calculate the following integral: $\displaystyle \int \dfrac {x^{5}-x}{x^{8}+1} dx$ .

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Problem 1920

Using various methods, calculate the following integral: $\displaystyle \int \dfrac {x^{4}+1}{x^{6}+1} dx$ .

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Problem 1921

Derive a reduction formula for computing the integral $\displaystyle I_{n}=\int \dfrac {dx}{(ax^{2}+bx+c)^{n}} (a\neq 0)$ .

Using this formula, compute $\displaystyle I_{3}=\int \dfrac {dx}{(x^{2}+x+1)^{3}}$ .

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Problem 1922

Using the substitution $t=\dfrac {x+a}{x+b}$ , compute the integral:

I=\int \dfrac {dx}{(x+a)^{m}(x+b)^{n}}

( $m$ and $n$ are natural numbers).

Using this substitution, find $\displaystyle \int \dfrac {dx}{(x-2)^{2}(x+3)^{3}}$ .

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Problem 1923

If $P_{n}(x)$ is a polynomial of degree $n$ in $x$ , compute $\displaystyle \int \dfrac {P_{n}(x)}{(x-a)^{n+1}} dx$ .

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Problem 1924

Let $R(x)=R^{*}(x^{2})$ , where $R^{*}$ is a rational function. What special properties does the function $R(x)$ have when it is decomposed into partial fractions?

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Problem 1925

Compute $\displaystyle \int \dfrac {dx}{1+x^{2n}}$ , where $n$ is a positive integer.

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