Questions tagged [reduction-formula]

This tag is for those who are trying to prove or derive reduction formulas of integrals. Reduction formulas are often useful to those trying to integrate trigonometric, exponential, or rational functions raised to certain powers, or functions containing multiple variables.

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Reduction formula Integration

Is it true that $I_{p,q}=\left ( \frac{p-1}{p+q} \right )I_{p-q,q}$ given that $I_{p,q}=\int_{0}^{\frac{\pi}{2}}\sin^{p}x \cos x dx$ (where $p$ and $q$ are positive integers)? If so, how? Also, how ...
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Design FPT algorithm for Defendable set

Design an FPT algorithm for Defendable Set parameterized by p: = n - k, where n = |V | is the number of vertices of D. It is recommended to structure the solution as follows. Consider a partition of ...
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1answer
28 views

Proving an inequality of reduction formula by induction

If $I_n = \int_0^1 t^n e^{-t} dt$ for $n \geq 0$ then $I_n=nI_{n-1} - e^{-1}$ for all $n \geq 1$. Prove by induction that, for all positive integers $n$, $I_n < n!$ How do I prove this ??
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84 views

Recurrence relation for $\int_{0}^{\infty} \frac{1}{(1+x^2/a)^n}dx$

Let$$I_{n,a} = \int_{0}^{\infty} \frac{1}{(1+x^2/a)^n}dx$$ where $a>0$. Show that $$I_{n+1,a} = \frac{2n-1}{2n}I_{n,a}$$ I have tried integrating by parts but it didn't work for me, and I don't ...
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46 views

Deriving the reduction formula for $\int\cos^n x\,\mathrm{d}x$

When they add $(n-1)\int \cos^{n}xdx$ to both sides of the equation how does $-(n-1)\int \cos^{n}xdx$ become $n\int \cos^{n}xdx$? Shouldn't it become $(n-1)\int \cos^{n}xdx$ on the left and the one on ...
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46 views

Help solving an improper integral with recursion of some sort

So i have this integral: $$\int_{0}^{\infty} e^{-x}\sin^nxdx$$ So I am not allowed to use the reduction formulae as a fact, without giving a proof, well maybe for $sin^n x$ only, but okay. My first ...
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36 views

Recurrence relation for integral trouble

For integer $n \ge 0$ define $$ F_n(x) = \int \frac{x^{n + 2}}{\sqrt{x^3 + 1}}~{\rm d}x $$ Find a recurrence relation between $F_n(x)$ and $F_{n-3}(x)$ and hence calculate $F_3(x)$ and $F_6(x)$ ...
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77 views

Reduction formula for definite integral of $\sin^2(x)$.

When trying to proof the reduction formula: $$I_n =\frac{n-1}{n}\cdot I_{n-2}$$ for the definite integral $I_n:=\int_{0}^{\pi} \sin(x)^{n} dx$, I tried going about it by reducing the indefinite ...
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2answers
75 views

Recursional Formula for Integration

Consider the following integral, $$I(n)=\int_0^{\pi/2}\cos^nx\cos(nx)dx $$ I tried taking one $\cos x$ out and then integrating by parts. I also tried integrating by parts using $\cos(nx)$ as the ...
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65 views

Reduction Formulae Integral $x(1-x^3)$

The question asks us: "If $$u_n=\int_0^1x(1-x^3)^ndx$$ show that $$u_n= \frac {3n}{3n+2}u_{n-1}$$ I've tried integration by parts using a coefficient of $1, x$ and even tried reducing the $...
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329 views

Given $ I_n=\int_{0}^{1}\frac{(1-x)^n}{n!}e^x\,dx $, prove that $ I_n=\frac{1}{(n+1)!}+I_{n+1} $

$$ I_n=\int_{0}^{1}\frac{(1-x)^n}{n!}e^x\,dx $$ Prove that $$ I_n=\frac{1}{(n+1)!}+I_{n+1} $$ I tried integration by parts and still can't prove it, I appreciate any hint/answer.
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134 views

By creating a reduction formula for $I_n=\int_0 ^\pi x^n \sin x \,dx$ Show that $I_4=\pi^4 -12\pi^2 + 48$

By creating a reduction formula for $$I_n=\int_0 ^\pi x^n \sin x \,dx$$ Show that $I_4=\pi^4 -12\pi^2 + 48$ So by using integration by parts I wrote $I_n$ as $$I_n=\int_0 ^\pi x^n \sin x \,dx = \...
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5answers
113 views

Evaluate $\int_{0}^{\pi/2}\cos^{2n+1}(x)dx$

How can I compute this Integral for integer $n$ from $0$ to $\pi/2$ $$\int_{0}^{\pi/2}\cos^{2n+1}(x)dx?$$
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76 views

Determine the recurrence formula

$\int_{-1}^{1}(1-x^2)^ndx$ I have trouble with finding recurrence formula for this integral. $n$ is natural parameter. I've tried to split up $(1-x^2)^n = (1+x)^n(1-x)^n$ and then to integrate ...
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2answers
74 views

General recurrence $f(n)=\alpha(n)+\beta(n)f(n-1)$

While computing certain integrals, like $$I_n=\int\frac{\mathrm dx}{(ax^2+b)^{n+1}}$$ I frequently come up with recurrence relations (AKA reduction formulae) like $$I_n=\frac{x}{2bn(ax^2+b)^n}+\frac{...
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1answer
96 views

$I_{m,n}=\int\frac{x^m}{(ax^2+bx+c)^n}dx$ Reduction Formula

I'm having trouble proving the following reduction formula: If $$I_{m,n}=\int\frac{x^m}{(ax^2+bx+c)^n}dx$$ then $$\int\frac{x^m}{(ax^2+bx+c)^n}dx=-\frac{x^{m-1}}{a(2m-n-1)(ax^2+bx+c)^{n-1}}-\frac{...
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2answers
106 views

Proving reduction formula using integration by parts

I'm having difficulty proving this integral reduction formula by parts If $$I_n=\int\frac{dx}{(a^2-x^2)^n}$$ , then $$\int\frac{dx}{(a^2-x^2)^n}=\frac{x}{2a^2(n-1)(a^2-x^2)^{n-1}}+\left(\frac{2n-...
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1answer
94 views

Integral $\int \frac{\sin^n(x)}{\cos(x)}dx$

In one of my exercises about integration we had to solve the following integral: \begin{equation} \int \frac{\sin^n(x)}{\cos^m(x)}dx \end{equation} We had to do this via a recursive integral. I ...
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198 views

Integration of $e^{ax}\cos^n bx$ and $e^{ax}\sin^n bx$

Q: Integration of $e^{ax}\cos^n bx$ and $e^{ax}\sin^n bx$ I know how to integration $e^{ax}\cos bx$ using $\cos bx=\frac{e^{ibx}+e^{-ibx}}{2}$.Using the same trick here I got $$\int e^{ax}\cos^n bx ~...
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82 views

Reduction formula for $\int\frac{dx}{(ax^2+b)^n}$

I recently stumbled upon the following reduction formula on the internet which I am so far unable to prove. $$I_n=\int\frac{\mathrm{d}x}{(ax^2+b)^n}\\I_n=\frac{x}{2b(n-1)(ax^2+b)^{n-1}}+\frac{2n-3}{2b(...
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2answers
66 views

Find a reduction formula for a trigonometric integral

$$\int\tan^{2n}(x)\sec^3(x)\,dx$$ I am aware that we use integration by parts to find the reduction formula, but I'm stuggling with what to use for $u,$ and $v.$ I've tried letting $$u=\tan^{2n-2}(x)$$...
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1answer
76 views

Reduction formula of $\int \left(\frac{x^{n-2}}{x^n-1}\right)dx$

I need help in finding reduction formula of the following: $$ I_n=\int \left(\frac{x^{n-2}}{x^n-1}\right)dx $$ Any hint or a complete solution would be very helpful.
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4answers
136 views

Integration reduction formula

How to create a reduction formula for the integral$$\int x \cos^n x \;\mathrm{d}x$$ I have tried everything i could think of, but I'm not even close to solving it. Really need help.
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66 views

Evaluating $\int \frac{x^n}{y}dx$ using reduction formula

If $y^2=3x^2+2x+1$ and integration $I_n$ is defined as $I_n= \int \frac{x^n}{y}dx$, where $AI_{10}+BI_{9}+CI_{8}=x^9y$, then find the values of $A,B,C$. I did generate the reduction formula but I am ...
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2answers
3k views

A reduction formula for $\int_0^1 x^n/\sqrt{9 - x^2}\,\mathrm dx$

Let $$I_n = \int_0^1 \frac{x^n}{\sqrt{9 - x^2}}\,\mathrm dx$$ Using integration, show that $$nI_n = 9(n - 1)nI_{n - 2} - 2\sqrt2$$ I've found that $\displaystyle I_0 = \sin^{-1}\left(\frac{1}{3}\...
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1answer
3k views

Proving the Reduction Formula of $\tan^n(\theta)$

The following is what I have tried to prove the reduction formula of $\tan^n(\theta)$. For the last step, how to convert the $\int sec^2(\theta) tan^{n-1}(\theta)d\theta $ to $\frac {tan^{n-1}\theta}{...
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1k views

Proof of Reduction Formula for $\int \cos^n (x) \ dx = \frac{1}{n}\cos^{n-1}x\sin x + \frac{n-1}{n}\int\cos^{n-2}x \ dx$ [duplicate]

I ran into a question with proving the reduction formula: $$ \int \cos^n x \ dx = \frac{1}{n}\cos^{n-1}x\sin x + \frac{n-1}{n}\int\cos^{n-2}x \ dx $$ I then attempted to prove by differentiation with ...
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4answers
128 views

Prove that $\int{\sec^{n}(\theta)}=\frac{\tan(\theta)\sec^{n-2}(\theta)}{n-1}-\frac{n-2}{n-1}\int{\sec^{n-2}(\theta)d\theta}$

One of the problems in my homework set ask me to prove the following identity: $$\int{\sec^{n}(\theta)}d\theta=\frac{\tan(\theta)\sec^{n-2}(\theta)}{n-1}-\frac{n-2}{n-1}\int{\sec^{n-2}(\theta)d\theta}$...
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1answer
919 views

Reduction Formula $ I_{(m,n)}\; =\;\int x^m(x+a)^ndx $

Some problem occured in proving the following reduction formula. $$ \\ I_{(m,n)}\; =\;\int x^m(x+a)^ndx\; = \; \frac{x^m(x+a)^{n+1}}{m+n+1}-\frac{ma}{m+n+1}I_{(m-1,n)}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;m,...
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2answers
26k views

Proving a reduction formula for the antiderivative of $\cos^n(x)$ [duplicate]

I want to show that for all $n\ge 2$, it holds that $$ \int \cos^n x\ dx = \frac{1}{n} \cos^{n-1} x \sin x + \frac{n-1}{n}\int \cos^{n-2} x\ dx. $$ I'm not even getting the result for the induction ...
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24k views

Prove $\int\cos^n x \ dx = \frac{1}n \cos^{n-1}x \sin x + \frac{n-1}{n}\int\cos^{n-2} x \ dx$

I am trying to prove $$\int\cos^n x \ dx = \frac{1}n \cos^{n-1}x \sin x + \frac{n-1}{n}\int\cos^{n-2} x \ dx$$ This problem is a classic, but I seem to be missing one step or the understanding of ...
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2answers
1k views

Reduction formula for $I_{n}=\int {\cos{nx} \over \cos{x}}\rm{d}x$

What would be a simple method to compute a reduction formula for the following? $\displaystyle I_{n}=\int {\cos{nx} \over \cos{x}} \rm{d}x~$ where $n$ is a positive integer I understand that it ...