Solve wrt x / ( x ^ 2 ) ^ 2 = ∫ 1 / ( 16 + x ^2)^2dx

$$\frac { d x } { ( x ^ { 2 } ) ^ { 2 } } = \int \frac { 1 } { ( 16 + x ^ { 2 } ) ^ { 2 } } d x$$

Solve for d (complex solution)

$$d=\frac{x^{3}\left(\sin(2\arctan(\frac{x}{4}))+2\arctan(\frac{x}{4})+256С\right)}{256}$$

$$x\neq 0\text{ and }x\neq -4i\text{ and }x\neq 4i$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x^{4}.

$$dx=x^{4}\int \frac{1}{\left(16+x^{2}\right)^{2}}\mathrm{d}x$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(16+x^{2}\right)^{2}.

$$dx=x^{4}\int \frac{1}{256+32x^{2}+\left(x^{2}\right)^{2}}\mathrm{d}x$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$dx=x^{4}\int \frac{1}{256+32x^{2}+x^{4}}\mathrm{d}x$$

The equation is in standard form.

$$xd=x^{4}\left(\frac{\sin(2\arctan(\frac{4}{x}))}{256}-\frac{\arctan(\frac{4}{x})}{128}+С\right)$$

Divide both sides by x.

$$\frac{xd}{x}=\frac{x^{4}\left(-\frac{\arctan(\frac{4}{x})}{128}+\frac{x}{32\left(x^{2}+16\right)}+С\right)}{x}$$

Dividing by x undoes the multiplication by x.

$$d=\frac{x^{4}\left(-\frac{\arctan(\frac{4}{x})}{128}+\frac{x}{32\left(x^{2}+16\right)}+С\right)}{x}$$

Divide x^{4}\left(-\frac{\arctan(\frac{4}{x})}{128}+\frac{x}{32\left(x^{2}+16\right)}+С\right) by x.

$$d=\frac{x^{3}\left(-x^{2}\arctan(\frac{4}{x})-16\arctan(\frac{4}{x})+128Сx^{2}+4x+2048С_{1}\right)}{128\left(x^{2}+16\right)}$$

Solve for d

$$d=\frac{x^{3}\left(x^{2}\arctan(\frac{x}{4})+16\arctan(\frac{x}{4})+С\left(x^{2}+1\right)+4x\right)}{128\left(x^{2}+16\right)}$$

$$x\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x^{4}.

$$dx=x^{4}\int \frac{1}{\left(16+x^{2}\right)^{2}}\mathrm{d}x$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(16+x^{2}\right)^{2}.

$$dx=x^{4}\int \frac{1}{256+32x^{2}+\left(x^{2}\right)^{2}}\mathrm{d}x$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$dx=x^{4}\int \frac{1}{256+32x^{2}+x^{4}}\mathrm{d}x$$

The equation is in standard form.

$$xd=x^{4}\left(\frac{\sin(2\arctan(\frac{4}{x}))}{256}-\frac{\arctan(\frac{4}{x})}{128}+С\right)$$

Divide both sides by x.

$$\frac{xd}{x}=\frac{x^{4}\left(-\frac{\arctan(\frac{4}{x})}{128}+\frac{x}{32\left(x^{2}+16\right)}+С\right)}{x}$$

Dividing by x undoes the multiplication by x.

$$d=\frac{x^{4}\left(-\frac{\arctan(\frac{4}{x})}{128}+\frac{x}{32\left(x^{2}+16\right)}+С\right)}{x}$$

Divide x^{4}\left(-\frac{\arctan(\frac{4}{x})}{128}+\frac{x}{32\left(x^{2}+16\right)}+С\right) by x.

$$d=\frac{x^{3}\left(-x^{2}\arctan(\frac{4}{x})-16\arctan(\frac{4}{x})+128Сx^{2}+4x+2048С_{1}\right)}{128\left(x^{2}+16\right)}$$