# 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 ...
9 views

### 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 ...
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 ??
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 ...
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 ...
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 ...
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 $F_6(x)$ ...
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 ...
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 ...