Solve (e^x-e^-x)sinx/x=f

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

Solve for f (complex solution)

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

$$x\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x.

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

Swap sides so that all variable terms are on the left hand side.

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

Use the distributive property to multiply e^{x}-e^{-x} by \sin(x).

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

The equation is in standard form.

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

Divide both sides by x.

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

Dividing by x undoes the multiplication by x.

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

Solve for f

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

$$x\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x.

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

Swap sides so that all variable terms are on the left hand side.

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

Use the distributive property to multiply e^{x}-e^{-x} by \sin(x).

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

The equation is in standard form.

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

Divide both sides by x.

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

Dividing by x undoes the multiplication by x.

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

Divide \sin(x)\left(e^{x}-\frac{1}{e^{x}}\right) by x.

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