What is the integral of (sqrt(x^4+1))/(x^3) ?

The integral of (sqrt(x^4+1))/(x^3) is -(sqrt(x^4+1))/(2x^2)+1/2 ln|sqrt(x^4+1)+x^2|+C

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integral of (sqrt(x^4+1))/(x^3)

\int \frac{x ^{4} + 1}{x ^{3}} d x

- \frac{x ^{4} + 1}{2 x ^{2}} + \frac{1}{2} ln x^{4} + 1 + x^{2} + C

Solution steps

\int \frac{x ^{4} + 1}{x ^{3}} d x

Apply Integration By Parts

: - \frac{x ^{4} + 1}{2 x ^{2}} - \int - \frac{x}{x ^{4} + 1} d x

= - \frac{x ^{4} + 1}{2 x ^{2}} - \int - \frac{x}{x ^{4} + 1} d x

\int - \frac{x}{x ^{4} + 1} d x = - \frac{1}{2} ln x^{4} + 1 + x^{2}

= - \frac{x ^{4} + 1}{2 x ^{2}} - (- \frac{1}{2} ln x^{4} + 1 + x^{2})

Simplify

= - \frac{x ^{4} + 1}{2 x ^{2}} + \frac{1}{2} ln x^{4} + 1 + x^{2}

Add a constant to the solution

= - \frac{x ^{4} + 1}{2 x ^{2}} + \frac{1}{2} ln x^{4} + 1 + x^{2} + C

x \to 3 lim (\frac{2}{( x - 3 ) ^{3}})

x \to 1 lim (\frac{17 - x ^{2} - 4}{x - 1})

\frac{x ^{2}}{18 x - 9}

\frac{d}{d x} (\frac{4}{5} x^{15})

What is the integral of (sqrt(x^4+1))/(x^3) ?
The integral of (sqrt(x^4+1))/(x^3) is -(sqrt(x^4+1))/(2x^2)+1/2 ln|sqrt(x^4+1)+x^2|+C