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Pattern - 33 of Indefinite Integration

February 20, 2021
Category: Pattern - 33 of Indefinite Integration

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⋆ Pattern – 33

$I_n=\ \int x^n sin\ mx\ dx, then I_n$ $=\ \frac{x^n\cos{mx}}{m}+\ \frac{nx^{n-1}\sin{mx}}{m^2}-\ \frac{n\left(n-1\right)}{m^2}I_{n-2}$ : n ≥ 2

Here, $\ I_n\ =\ \int\ x^n\ sin\ mx\ dx$

$\therefore\ \ I_n\ =\ x^n\ \int\ sin\ mx\ dx\ -\ \int\ \left(\frac{d}{dx}\ x^n\ \int s i n{mx}\ \ dx\right)dx$

$=\ \frac{-x^n\cos{mx}}{m}+\ \int n\ x^{n-1}\times\frac{\cos{mx}}{m}\ dx$

$=\ \frac{-x^n\cos{mx}}{m}+\frac{n}{m}\ \int x^{n-1}\cos{mx}\ dx$

$=\ \frac{-x^n\cos{mx}}{m}+\frac{n}{m}\ \left[x^{n-1}\ cos\int{mx}\ dx-\ \int\left(\frac{d}{dx}\ x^{n-1}\ cos\int{mx\ dx}\right)dx\right]$

$=\ \frac{-x^n\cos{mx}}{m}+\frac{n}{m}\ \left[x^{n-1\ }\times\frac{\sin{mx}}{m}-\int\left(n-1\right)x^{n-2}\times\frac{\sin{mx}}{m}\ dx\right]$

$=\ \frac{-x^n\cos{mx}}{m}+\ \frac{nx^{n-1}\sin{mx}}{m^2}-\ \frac{n\left(n-1\right)}{m^2}\int x^{n-2}\sin{mx}\ dx$

$=\ \frac{-x^n\cos{mx}}{m}+\ \frac{nx^{n-1}\sin{mx}}{m^2}-\ \frac{n\left(n-1\right)}{m^2}\ I_{n-2}\ ;n-2\geq0\ \therefore\ \ n\geq2$

1) Evaluate : $\int x^5\sin{3x}\ dx$

Solution. Here, m = 3 &

$I_5=\ \int x^5\sin{3x}\ dx$

$=\ \frac{x^5\cos{3x}}{3}+\ \frac{5x^4\sin{3x}}{9}-\ \frac{5\times4}{9}I_3$

$=\ \frac{-x^5\cos{3x}}{3}+\ \frac{5x^4\sin{3x}}{9}-\frac{20}{9}\left[\frac{-x^3\cos{3x}}{3}+\frac{3x^2\sin{3x}}{93}-\frac{6}{9}I_1\right]$

$=\ \frac{-x^5\cos{3x}}{3}+\ \frac{5x^4\sin{3x}}{9}+\frac{20}{27}\ x^3\cos{3x}-\ \frac{20x^2\sin{3x}}{27}+\frac{40}{27}\ I_1$

$=\ \frac{-x^5\cos{3x}}{3}+\ \frac{5x^4\sin{3x}}{9}+\frac{20x^3\cos{3x}}{27}-\frac{20x^2\sin{3x}}{27}+\frac{40}{27}\ I_1--(1)$

where,

$I_1$ = ∫ sin 3x dx

= x ∫ sin 3x dx – ∫ $\left(\frac{d}{dx}\ x\ sin\int{\sin{3x}\ dx}\right)dx$

$=\ \frac{-x\cos{3x}}{3}+\ \int\times\ \frac{\cos{3x}}{3}\ dx$

$=\ \frac{-x\cos{3x}}{3}+\frac{1}{3}\times\ \frac{\sin{3x}}{3}$

$I_5=\ \frac{-x^5\cos{3x}}{3}+\ \frac{5x^4\sin{3x}}{9}+\frac{20x^2\cos{3x}}{27}-\frac{20x^2\sin{3x}}{27}-\ \frac{40x\cos{3x}}{81}+\ \frac{40\sin{3x}}{243}+C$

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Pattern - 33 of Indefinite Integration

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