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Popular >

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

Solution

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

Solution

\frac{1}{2} arctan (x^{2}) + 3 (\frac{1}{4 2} (ln 2 x^{2} + 22 x + 2 + 2 arctan (2 x + 1)) + \frac{1}{4 2} (- ln 2 x^{2} - 22 x + 2 + 2 arctan (2 x - 1))) + C

Solution steps

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

Expand

\frac{x + 3}{x ^{4} + 1} : \frac{x}{x ^{4} + 1} + \frac{3}{x ^{4} + 1}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int \frac{x}{x ^{4} + 1} d x = \frac{1}{2} arctan (x^{2})

\int \frac{3}{x ^{4} + 1} d x = 3 (\frac{1}{4 2} (ln 2 x^{2} + 22 x + 2 + 2 arctan (2 x + 1)) + \frac{1}{4 2} (- ln 2 x^{2} - 22 x + 2 + 2 arctan (2 x - 1)))

= \frac{1}{2} arctan (x^{2}) + 3 (\frac{1}{4 2} (ln 2 x^{2} + 22 x + 2 + 2 arctan (2 x + 1)) + \frac{1}{4 2} (- ln 2 x^{2} - 22 x + 2 + 2 arctan (2 x - 1)))

Add a constant to the solution

= \frac{1}{2} arctan (x^{2}) + 3 (\frac{1}{4 2} (ln 2 x^{2} + 22 x + 2 + 2 arctan (2 x + 1)) + \frac{1}{4 2} (- ln 2 x^{2} - 22 x + 2 + 2 arctan (2 x - 1))) + C

Popular Examples

integral of (e^t)/(4+e^t)

\int \frac{e ^{t}}{4 + e ^{t}} d t

integral of ((t+9)^2)/(t^2)

\int \frac{( t + 9 ) ^{2}}{t ^{2}} d t

integral of sec^2(x)cos^2(x)

\int sec^{2} (x) cos^{2} (x) d x

integral of sin(5\times 1+1)

\int sin (5 \times 1 + 1) d x

integral of x^3*ln(x^2)

\int x^{3} \cdot ln (x^{2}) d x

Frequently Asked Questions (FAQ)

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