Question

If $I_m=\int \frac{x^m}{\sqrt{ax^2+c}}\, dx$, show that $amI_m=x^{m-1}\sqrt{ax^2+c}-(m-1)c I_{m-2}$.

score 38 · Answer 1 · 2026-03-25 23:42:28

38

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If $I_m=\int \frac{x^m}{\sqrt{ax^2+c}}\, dx$, show that $amI_m=x^{m-1}\sqrt{ax^2+c}-(m-1)c I_{m-2}$.

Published on 25 Mar 2026 - 23:42

score 11.4k · Answer 2 · 2026-03-25 19:10:31

11.4k

Views

Prove the reduction formula for $\int\limits_{0}^{\pi/2} \sin^n x\, dx=\int\limits_{0}^{\pi/2} \cos^n x\, dx$

Published on 25 Mar 2026 - 19:10

score 77 · Answer 3 · 2026-04-15 18:54:33

77

Views

Prove by increasing the inequalities $\frac{1-\sin^{2n-1}x}{2n-1}> \frac{1-\sin^{2n}x}{2n}>\frac{1-\sin^{2n+1}x}{2n+1}$

Published on 15 Apr 2026 - 18:54

score 226 · Answer 4 · 2026-04-16 14:51:23

226

Views

Reduction formula for $I_{m,n}=\int x^m \mathrm{cosec}^n x\; dx$.

Published on 16 Apr 2026 - 14:51

score 24 · Answer 5 · 2026-04-09 07:20:14

24

Views

How to calculate max affordability

Published on 09 Apr 2026 - 7:20

score 208 · Answer 6 · 2026-04-11 08:15:12

208

Views

How to get a reduction formula for this integral?

Published on 11 Apr 2026 - 8:15

score 62 · Answer 7 · 2026-04-13 01:59:46

62

Views

Alternative Reduction Formula for $\int\cos^n{x}\,dx$

Published on 13 Apr 2026 - 1:59

score 1.1k · Answer 8 · 2026-04-17 10:14:17

1.1k

Views

Proof that $\int_0^{\pi/2}\cos^m(x)\cos (nx)\,dx=\frac m{m+n}\int_0^{\pi/2}\cos^{m-1}(x)\cos((n-1)x)\,dx$

Published on 17 Apr 2026 - 10:14

score 53 · Answer 9 · 2026-04-16 07:54:09

53

Views

Given $I_n = \int_0^1 \frac{x^n}{\sqrt{x^3+1}} dx$, show $(2n-1)I_n+2(n-2)I_{n-3}=2\sqrt{2}$

Published on 16 Apr 2026 - 7:54

score 318 · Answer 10 · 2026-04-16 03:01:59

318

Views

Deriving a reduction formula for the integral: $I_n=\int _0^1 \left(1+x^2\right)^n \mathrm{d}x$

Published on 16 Apr 2026 - 3:01

score 2.2k · Answer 11 · 2026-04-16 02:35:05

2.2k

Views

Prove formula for $\int \frac{dx}{(1+x^2)^n}$

Published on 16 Apr 2026 - 2:35

score 133 · Answer 12 · 2026-04-15 13:01:42

133

Views

Show $n I_n I_{n-1} = \frac{\pi}{2}$ where $I_n = \displaystyle\int_0^{\pi/2} \cos^n x dx$.

Published on 15 Apr 2026 - 13:01

score 80 · Answer 13 · 2026-04-15 20:53:34

80

Views

Show that $n\int_0^{2\pi}\sin^n\theta d \theta = (n-1)\int_0^{2\pi}\sin^{(n-2)}\theta d\theta$

Published on 15 Apr 2026 - 20:53

score 299 · Answer 14 · 2026-04-30 08:38:09

299

Views

Find a Reduction Formula

Published on 30 Apr 2026 - 8:38

score 320 · Answer 15 · 2026-05-10 12:55:49

320

Views

Integration reduction formula

Published on 10 May 2026 - 12:55

List Question

If $I_m=\int \frac{x^m}{\sqrt{ax^2+c}}\, dx$, show that $amI_m=x^{m-1}\sqrt{ax^2+c}-(m-1)c I_{m-2}$.

Prove the reduction formula for $\int\limits_{0}^{\pi/2} \sin^n x\, dx=\int\limits_{0}^{\pi/2} \cos^n x\, dx$

Prove by increasing the inequalities $\frac{1-\sin^{2n-1}x}{2n-1}> \frac{1-\sin^{2n}x}{2n}>\frac{1-\sin^{2n+1}x}{2n+1}$

Reduction formula for $I_{m,n}=\int x^m \mathrm{cosec}^n x\; dx$.

How to calculate max affordability

How to get a reduction formula for this integral?

Alternative Reduction Formula for $\int\cos^n{x}\,dx$

Proof that $\int_0^{\pi/2}\cos^m(x)\cos (nx)\,dx=\frac m{m+n}\int_0^{\pi/2}\cos^{m-1}(x)\cos((n-1)x)\,dx$

Given $I_n = \int_0^1 \frac{x^n}{\sqrt{x^3+1}} dx$, show $(2n-1)I_n+2(n-2)I_{n-3}=2\sqrt{2}$

Deriving a reduction formula for the integral: $I_n=\int _0^1 \left(1+x^2\right)^n \mathrm{d}x$

Prove formula for $\int \frac{dx}{(1+x^2)^n}$

Show $n I_n I_{n-1} = \frac{\pi}{2}$ where $I_n = \displaystyle\int_0^{\pi/2} \cos^n x dx$.

Show that $n\int_0^{2\pi}\sin^n\theta d \theta = (n-1)\int_0^{2\pi}\sin^{(n-2)}\theta d\theta$

Find a Reduction Formula

Integration reduction formula

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