Whakaoti i te (e^x-e^-x)sinx/x=f

$$\frac { ( e ^ { x } - e ^ { - x } ) \sin x } { x } = f$$

Whakaoti mō f (complex solution)

$$f=\frac{ie^{\left(2-i\right)x}-ie^{\left(2+i\right)x}-2\sin(x)}{2xe^{x}}$$

$$x\neq 0$$

Ngā Upane mō te Whakaoti Whārite Rārangi

Whakareatia ngā taha e rua o te whārite ki te x.

$$\left(e^{x}-e^{-x}\right)\sin(x)=fx$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$fx=\left(e^{x}-e^{-x}\right)\sin(x)$$

Whakamahia te āhuatanga tohatoha hei whakarea te e^{x}-e^{-x} ki te \sin(x).

$$fx=e^{x}\sin(x)-e^{-x}\sin(x)$$

He hanga arowhānui tō te whārite.

$$xf=\sin(x)e^{x}-\frac{\sin(x)}{e^{x}}$$

Whakawehea ngā taha e rua ki te x.

$$\frac{xf}{x}=\frac{\sin(x)\left(-\frac{1}{e^{x}}+e^{x}\right)}{x}$$

Mā te whakawehe ki te x ka wetekia te whakareanga ki te x.

$$f=\frac{\sin(x)\left(-\frac{1}{e^{x}}+e^{x}\right)}{x}$$

Whakaoti mō f

$$f=\frac{\sin(x)\left(e^{2x}-1\right)}{xe^{x}}$$

$$x\neq 0$$

Ngā Upane mō te Whakaoti Whārite Rārangi

Whakareatia ngā taha e rua o te whārite ki te x.

$$\left(e^{x}-e^{-x}\right)\sin(x)=fx$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$fx=\left(e^{x}-e^{-x}\right)\sin(x)$$

Whakamahia te āhuatanga tohatoha hei whakarea te e^{x}-e^{-x} ki te \sin(x).

$$fx=e^{x}\sin(x)-e^{-x}\sin(x)$$

He hanga arowhānui tō te whārite.

$$xf=\sin(x)e^{x}-\frac{\sin(x)}{e^{x}}$$

Whakawehea ngā taha e rua ki te x.

$$\frac{xf}{x}=\frac{\sin(x)\left(-\frac{1}{e^{x}}+e^{x}\right)}{x}$$

Mā te whakawehe ki te x ka wetekia te whakareanga ki te x.

$$f=\frac{\sin(x)\left(-\frac{1}{e^{x}}+e^{x}\right)}{x}$$

Whakawehe \sin(x)\left(e^{x}-\frac{1}{e^{x}}\right) ki te x.

$$f=\frac{\sin(x)\left(e^{2x}-1\right)}{xe^{x}}$$