Math Problems Vault

Problem 1991

Compute using the recursion formula. Find the following integral: $\displaystyle \int \cos^{5}x\, dx$ .

?

Problem 1992

Find the integral: $\displaystyle \int \sin^{6}x\, dx$ .

?

Problem 1993

Find the integral: $\displaystyle \int \cos^{6}x\, dx$ .

?

Problem 1994

Find the integral: $\displaystyle \int \sin^{2}x\cos^{4}x\, dx$ .

?

Problem 1995

Find the integral: $\displaystyle \int \sin^{4}x\cos^{5}x\, dx$ .

?

Problem 1996

Find the integral: $\displaystyle \int \sin^{5}x\cos^{5}x\, dx$ .

?

Problem 1997

Find the integral: $\displaystyle \int \dfrac {\sin^{3}x}{\cos^{4}x}\, dx$ .

?

Problem 1998

Find the integral: $\displaystyle \int \dfrac {\cos^{4}x}{\sin^{3}x}\, dx$ .

?

Problem 1999

Find the integral: $\displaystyle \int \dfrac {dx}{\sin^{3}x}$ .

?

Problem 2000

Find the integral: $\displaystyle \int \dfrac {dx}{\cos^{3}x}$ .

?

Problem 2001

Find the integral: $\displaystyle \int \dfrac {dx}{\sin^{4}x\cos^{4}x}$ .

?

Problem 2002

Find the integral: $\displaystyle \int \dfrac {dx}{\sin^{3}x\cos^{5}x}$ .

?

Problem 2003

Find the integral: $\displaystyle \int \dfrac {dx}{\sin x\cos^{4}x}$ .

?

Problem 2004

Find the integral: $\displaystyle \int \operatorname {tg}^{5}x\, dx$ .

?

Problem 2005

Find the integral: $\displaystyle \int \operatorname {ctg}^{6}x\, dx$ .

?

Problem 2006

Find the integral: $\displaystyle \int \dfrac {\sin^{4}x}{\cos^{8}x}\, dx$ .

?

Problem 2007

Find the integral: $\displaystyle \int \dfrac {dx}{\sqrt{\sin^{3}x\cos^{5}x}}$ .

?

Problem 2008

Find the integral: $\displaystyle \int \dfrac {dx}{\cos x\sqrt[3]{\sin^{2}x}}$ .

?

Problem 2009

Find the integral: $\displaystyle \int \dfrac {dx}{\sqrt{\operatorname {tg}x}}$ .

?

Problem 2010

Find the integral: $\displaystyle \int \dfrac {dx}{\sqrt[3]{\operatorname {tg}x}}$ .

?

Problem 2011

(a)(b)

Derive recursion formulas for the following integrals:

(1) $I_{n}=\displaystyle \int \sin^{n}x\, dx$ ；

(2) $K_{n}=\displaystyle \int \cos^{n}x\, dx$ ( $n>2$ ). and use the resulting formulas to compute $\displaystyle \int \sin^{6}x\, dx$ and $\displaystyle \int \cos^{8}x\, dx$ .

?

(a)

(b)

Problem 2012

(a)(b)

Derive recursion formulas for the following integrals:

(1) $I_{n}=\displaystyle \int \dfrac {dx}{\sin^{n}x}$ ；

(2) $K_{n}=\displaystyle \int \dfrac {dx}{\cos^{n}x}$ ( $n>2$ ). and use the resulting formulas to compute $\displaystyle \int \dfrac {dx}{\sin^{5}x}$ and $\displaystyle \int \dfrac {dx}{\cos^{7}x}$ .

?

(a)

(b)

Problem 2013

Using the formulas

\text{I}\quad \sin \alpha \sin \beta =\frac{1}{2}\left[\cos (\alpha -\beta )-\cos (\alpha +\beta )\right] ;

\text{II}\quad \cos \alpha \cos \beta =\frac{1}{2}\left[\cos (\alpha +\beta )+\cos (\alpha -\beta )\right] ;

\text{III}\quad \sin \alpha \cos \beta =\frac{1}{2}\left[\sin (\alpha -\beta )+\sin (\alpha +\beta )\right]

to compute the following integrals. Find the integral: $\displaystyle \int \sin 5x\cos x\, dx$ .

?

Problem 2014

Find the integral: $\displaystyle \int \cos x\cos 2x\cos 3x\, dx$ .

?

Problem 2015

Find the integral: $\displaystyle \int \sin x\sin \dfrac {x}{2}\sin \dfrac {x}{3}\, dx$ .

?

Problem 2016

Find the integral: $\displaystyle \int \sin x\sin (x+a)\sin (x+b)\, dx$ .

?

Problem 2017

Find the integral: $\displaystyle \int \cos^{2}ax\cos^{2}bx\, dx$ .

?

Problem 2018

Find the integral: $\displaystyle \int \sin^{3}2x\cos^{2}3x\, dx$ .

?

Problem 2019

Find the integral: $\displaystyle \int \dfrac {dx}{\sin (x+a)\sin (x+b)}$ .

?

Problem 2020

Find the integral: $\displaystyle \int \dfrac {dx}{\sin (x+a)\cos (x+b)}$ .

?

Problem 2021

Find the integral: $\displaystyle \int \dfrac {dx}{\cos (x+a)\cos (x+b)}$ .

?

Problem 2022

Find the integral: $\displaystyle \int \dfrac {dx}{\sin x-\sin a}$ .

?

Problem 2023

Find the integral: $\displaystyle \int \dfrac {dx}{\cos x+\cos a}$ .

?

Problem 2024

Find the integral: $\displaystyle \int \operatorname {tg}x\operatorname {tg}(x+a)\, dx$ .

?

Problem 2025

(a)(b)

For an integral of the form $\displaystyle \int R(\sin x,\cos x)\, dx$ (where $R$ is a rational function), the general case can be reduced to the integral of a rational function by the substitution $\operatorname {tg}\dfrac {x}{2}=t$ .

?

(a)

If the identity $R(-\sin x,\cos x)\equiv -R(\sin x,\cos x)$ holds, or $R(\sin x,-\cos x)\equiv -R(\sin x,\cos x)$ holds, then it is best to use the substitution $\cos x=t$ or, correspondingly, $\sin x=t$ .

(b)

If the identity $R(-\sin x,-\cos x)\equiv R(\sin x,\cos x)$ holds, then it is best to use the substitution $\operatorname {tg}x=t$ .

Find the integral: $\displaystyle \int \dfrac {dx}{2\sin x-\cos x+5}$ .

Problem 2026

Find the integral: $\displaystyle \int \dfrac {dx}{(2+\cos x)\sin x}$ .

?

Problem 2027

Find the integral: $\displaystyle \int \dfrac {\sin^{2}x}{\sin x+2\cos x}\, dx$ .

?

Problem 2028

(a)(b)

Find the integral: $\displaystyle \int \dfrac {dx}{1+\varepsilon \cos x}$ :

(1) $0<\varepsilon <1$ ;

(2) $\varepsilon >1$ .

?

(a)

(b)

Problem 2029

Find the integral: $\displaystyle \int \dfrac {\sin^{2}x}{1+\sin^{2}x}\, dx$ .

?

Problem 2030

Find the integral: $\displaystyle \int \dfrac {dx}{a^{2}\sin^{2}x+b^{2}\cos^{2}x}$ .

?

Problem 2031

Find the integral: $\displaystyle \int \dfrac {\cos^{2}x\, dx}{(a^{2}\sin^{2}x+b^{2}\cos^{2}x)^{2}}$ .

?

Problem 2032

Find the integral: $\displaystyle \int \dfrac {\sin x\cos x}{\sin x+\cos x}\, dx$ .

?

Problem 2033

Find the integral: $\displaystyle \int \dfrac {dx}{(a\sin x+b\cos x)^{2}}$ .

?

Problem 2034

Find the integral: $\displaystyle \int \dfrac {\sin x\, dx}{\sin^{3}x+\cos^{3}x}$ .

?

Problem 2035

Find the integral: $\displaystyle \int \dfrac {dx}{\sin^{4}x+\cos^{4}x}$ .

?

Problem 2036

Find the integral: $\displaystyle \int \dfrac {\sin^{2}x\cos^{2}x}{\sin^{8}x+\cos^{8}x}\, dx$ .

?

Problem 2037

Find the integral: $\displaystyle \int \dfrac {\sin^{2}x-\cos^{2}x}{\sin^{4}x+\cos^{4}x}\, dx$ .

?

Problem 2038

Find the integral: $\displaystyle \int \dfrac {\sin x\cos x}{1+\sin^{4}x}\, dx$ .

?

Problem 2039

Find the integral: $\displaystyle \int \dfrac {dx}{\sin^{6}x+\cos^{6}x}$ .

?

Problem 2040

Find the integral: $\displaystyle \int \dfrac {dx}{(\sin^{2}x+2\cos^{2}x)^{2}}$ .

?

Problem 2041

Find the integral: $\displaystyle \int \dfrac {dx}{a\sin x+b\cos x}$ (first reduce the denominator to the form of a logarithm).

?

Problem 2042

Prove that $\displaystyle \int \dfrac {a_{1}\sin x+b_{1}\cos x}{a\sin x+b\cos x}\, dx=Ax+B\ln \left|a\sin x+b\cos x\right|+C$ , where $A$ , $B$ , $C$ are constants.

?

Problem 2043

Find the integral: $\displaystyle \int \dfrac {\sin x-\cos x}{\sin x+2\cos x}\, dx$ .

?

Problem 2043.1

[NO TASK]

Problem 2044

Find the integral: $\displaystyle \int \dfrac {dx}{3+5\operatorname {tg}x}$ .

?

Problem 2045

Find the integral: $\displaystyle \int \dfrac {a_{1}\sin x+b_{1}\cos x}{(a\sin x+b\cos x)^{2}}\, dx$ .

?

Problem 2046

Prove that: $\displaystyle \int \dfrac {a_{1}\sin x+b_{1}\cos x+c_{1}}{a\sin x+b\cos x+c}\, dx=Ax+B\ln \left|a\sin x+b\cos x+c\right|+C\displaystyle \int \dfrac {dx}{a\sin x+b\cos x+c}$ , where $A$ , $B$ , $C$ are all constant coefficients.

?

Problem 2047

Find the integral: $\displaystyle \int \dfrac {\sin x+2\cos x-3}{\sin x-2\cos x+3}\, dx$ .

?

Problem 2048

Find the integral: $\displaystyle \int \dfrac {\sin x\, dx}{\sqrt{2}+\sin x+\cos x}$ .

?

Problem 2049

Find the integral: $\displaystyle \int \dfrac {2\sin x+\cos x}{3\sin x+4\cos x-2}\, dx$ .

?

Problem 2050

Prove that: $\displaystyle \int \dfrac {a_{1}\sin^{2}x+2b_{1}\sin x\cos x+c_{1}\cos^{2}x}{a\sin x+b\cos x}\, dx=A\sin x+B\cos x+C\displaystyle \int \dfrac {dx}{a\sin x+b\cos x}$ , where $A$ , $B$ , $C$ are all constant coefficients.

?

Problem 2051

Find the integral: $\displaystyle \int \dfrac {\sin^{2}x-4\sin x\cos x+3\cos^{2}x}{\sin x+\cos x}\, dx$ .

?

Problem 2052

Find the integral: $\displaystyle \int \dfrac {\sin^{2}x-\sin x\cos x+2\cos^{2}x}{\sin x+2\cos x}\, dx$ .

?

Problem 2053

Prove that if $(a-c)^{2}+b^{2}\neq 0$ , then

\int \dfrac {a_{1}\sin x+b_{1}\cos x}{a\sin ^{2}x+2b\sin x\cos x+c\cos ^{2}x}\, dx=A\int \dfrac {du_{1}}{k_{1}u_{1}^{2}+\lambda _{1}}+B\int \dfrac {du_{2}}{k_{2}u_{2}^{2}+\lambda _{2}} ,

where $A$ , $B$ are undetermined coefficients, $\lambda_{1}$ , $\lambda_{2}$ are the roots of the equation below

\begin{vmatrix} a-\lambda & b \\ b & c-\lambda \end{vmatrix}=0\quad (\lambda _{1}\neq \lambda _{2})

and

u_{i}=(a-\lambda _{i})\sin x+b\cos x ，\quad k_{i}=\dfrac {1}{a-\lambda _{i}}\quad (i=1,2) .

?

Problem 2054

Find the integral: $\displaystyle \int \dfrac {2\sin x-\cos x}{3\sin^{2}x+4\cos^{2}x}\, dx$ .

?

Problem 2055

Find the integral: $\displaystyle \int \dfrac {\sin x+\cos x}{2\sin^{2}x-4\sin x\cos x+5\cos^{2}x}\, dx$ .

?

Problem 2056

Evaluate the integral: $\displaystyle \int \dfrac {\sin x-2\cos x}{1+4\sin x\cos x}\, dx$ .

?

Problem 2057

Prove that

\int \dfrac {dx}{(a\sin x+b\cos x)^{n}}=\dfrac {A\sin x+B\cos x}{(a\sin x+b\cos x)^{n-1}}+C\int \dfrac {dx}{(a\sin x+b\cos x)^{n-2}} ,

where $A$ , $B$ , $C$ are undetermined coefficients.

?

Problem 2058

Evaluate $\displaystyle \int \dfrac {dx}{(\sin x+2\cos x)^{3}}$ .

?

Problem 2059

If $n$ is a natural number greater than $1$ , prove that

\int \dfrac {dx}{(a+b\cos x)^{n}}=\dfrac {A\sin x}{(a+b\cos x)^{n-1}}+B\int \dfrac {dx}{(a+b\cos x)^{n-1}}+C\int \dfrac {dx}{(a+b\cos x)^{n-2}}\quad (\left|a\right|\neq \left|b\right|) ,

and find the coefficients $A$ , $B$ and $C$ .

?

Problem 2060

Evaluate the integral: $\displaystyle \int \dfrac {\sin x\, dx}{\cos x\sqrt{1+\sin^{2}x}}$ .

?

Problem 2061

Evaluate the integral: $\displaystyle \int \dfrac {\sin^{2}x}{\cos^{2}x\sqrt{\operatorname {tg}x}}\, dx$ .

?

Problem 2062

Evaluate the integral: $\displaystyle \int \dfrac {\sin x\, dx}{\sqrt{2+\sin 2x}}$ .

?

Problem 2063

Evaluate the integral: $\displaystyle \int \dfrac {dx}{(1+\varepsilon \cos x)^{2}}$ ( $0<\varepsilon <1$ ).

?

Problem 2064

Evaluate the integral: $\displaystyle \int \dfrac {\cos^{n-1}\dfrac {x+a}{2}}{\sin^{n+1}\dfrac {x-a}{2}}\, dx$ .

?

Problem 2065

Derive a recurrence formula for the integral $\displaystyle I_{n}=\int \left(\dfrac {\sin \dfrac {x-a}{2}}{\sin \dfrac {x+a}{2}}\right)^{n}\, dx$ ( $n$ a natural number).

?