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Identical expressions
- dx/(sixteen -x^ eight)
- dx divide by (16 minus x to the power of 8)
- dx divide by (sixteen minus x to the power of eight)
- dx/(16-x8)
- dx/16-x8
- dx/(16-x⁸)
- dx/16-x^8
- dx divide by (16-x^8)
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Similar expressions
- dx/(16+x^8)
Integral of dx/(16-x^8) dx
The solution
You have entered
[src]
1 / | | 1 | ------- dx | 8 | 16 - x | / 0
$$\int\limits_{0}^{1} \frac{1}{16 - x^{8}}\, dx$$
Integral(1/(16 - x^8), (x, 0, 1))
The answer (Indefinite)
[src]
/ / ___\
| ___ |x*\/ 2 |
|-\/ 2 *acoth|-------|
| \ 2 / 2
|---------------------- for x > 2
| 2
<
| / ___\
| ___ |x*\/ 2 |
|-\/ 2 *atanh|-------| / ___\
/ | \ 2 / 2 ___ |x*\/ 2 |
| |---------------------- for x < 2 / 2 \ / 2 \ \/ 2 *atan|-------|
| 1 \ 2 log\2 + x - 2*x/ atan(1 + x) atan(-1 + x) log\2 + x + 2*x/ \ 2 /
| ------- dx = C - ----------------------------------- - ----------------- + ----------- + ------------ + ----------------- + -------------------
| 8 32 128 64 64 128 64
| 16 - x
|
/
$$\int \frac{1}{16 - x^{8}}\, dx = C - \frac{\begin{cases} - \frac{\sqrt{2} \operatorname{acoth}{\left(\frac{\sqrt{2} x}{2} \right)}}{2} & \text{for}\: x^{2} > 2 \\- \frac{\sqrt{2} \operatorname{atanh}{\left(\frac{\sqrt{2} x}{2} \right)}}{2} & \text{for}\: x^{2} < 2 \end{cases}}{32} - \frac{\log{\left(x^{2} - 2 x + 2 \right)}}{128} + \frac{\log{\left(x^{2} + 2 x + 2 \right)}}{128} + \frac{\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} x}{2} \right)}}{64} + \frac{\operatorname{atan}{\left(x - 1 \right)}}{64} + \frac{\operatorname{atan}{\left(x + 1 \right)}}{64}$$
The graph
The answer
[src]
/ 1 I \ / 1 I \ ___ /pi*I / ___\\ ___ / pi*I / ___\\
___ / / ___\\ ___ / ___\ ___ / / ___\\ ___ / ___\ pi*I*|--- + ---| pi*I*|--- - ---| I*\/ 2 *|---- + log\\/ 2 /| ___ / ___\ I*\/ 2 *|- ---- + log\\/ 2 /| ___ / ___\
/ 1 I \ /pi*I / ___\\ / 1 I \ / pi*I / ___\\ / 1 I \ /3*pi*I / ___\\ / 1 I \ / 3*pi*I / ___\\ / 1 I \ / 1 I \ \/ 2 *\pi*I + log\-1 + \/ 2 // \/ 2 *log\\/ 2 / \/ 2 *\pi*I + log\\/ 2 // \/ 2 *log\1 + \/ 2 / \128 128/ \128 128/ \ 2 / I*\/ 2 *log\1 - I*\/ 2 / \ 2 / I*\/ 2 *log\1 + I*\/ 2 /
|- --- - ---|*|---- + log\\/ 2 /| + |- --- + ---|*|- ---- + log\\/ 2 /| + |--- - ---|*|------ + log\\/ 2 /| + |--- + ---|*|- ------ + log\\/ 2 /| - |- --- - ---|*log(2 + I) - |- --- + ---|*log(2 - I) - ------------------------------ - ---------------- + ------------------------- + -------------------- + ---------------- - ---------------- - --------------------------- - ------------------------ + ----------------------------- + ------------------------
\ 128 128/ \ 4 / \ 128 128/ \ 4 / \128 128/ \ 4 / \128 128/ \ 4 / \ 128 128/ \ 128 128/ 128 128 128 128 2 2 128 128 128 128
$$- \frac{\sqrt{2} \log{\left(\sqrt{2} \right)}}{128} + \frac{\sqrt{2} \log{\left(1 + \sqrt{2} \right)}}{128} - \frac{\sqrt{2} \left(\log{\left(-1 + \sqrt{2} \right)} + i \pi\right)}{128} + \left(\frac{1}{128} + \frac{i}{128}\right) \left(\log{\left(\sqrt{2} \right)} - \frac{3 i \pi}{4}\right) - \frac{i \pi \left(\frac{1}{128} - \frac{i}{128}\right)}{2} - \left(- \frac{1}{128} + \frac{i}{128}\right) \log{\left(2 - i \right)} + \left(- \frac{1}{128} - \frac{i}{128}\right) \left(\log{\left(\sqrt{2} \right)} + \frac{i \pi}{4}\right) - \frac{\sqrt{2} i \log{\left(1 - \sqrt{2} i \right)}}{128} - \frac{\sqrt{2} i \left(\log{\left(\sqrt{2} \right)} + \frac{i \pi}{2}\right)}{128} + \frac{\sqrt{2} i \left(\log{\left(\sqrt{2} \right)} - \frac{i \pi}{2}\right)}{128} + \frac{\sqrt{2} i \log{\left(1 + \sqrt{2} i \right)}}{128} + \left(- \frac{1}{128} + \frac{i}{128}\right) \left(\log{\left(\sqrt{2} \right)} - \frac{i \pi}{4}\right) - \left(- \frac{1}{128} - \frac{i}{128}\right) \log{\left(2 + i \right)} + \frac{i \pi \left(\frac{1}{128} + \frac{i}{128}\right)}{2} + \left(\frac{1}{128} - \frac{i}{128}\right) \left(\log{\left(\sqrt{2} \right)} + \frac{3 i \pi}{4}\right) + \frac{\sqrt{2} \left(\log{\left(\sqrt{2} \right)} + i \pi\right)}{128}$$
=
=
/ 1 I \ / 1 I \ ___ /pi*I / ___\\ ___ / pi*I / ___\\
___ / / ___\\ ___ / ___\ ___ / / ___\\ ___ / ___\ pi*I*|--- + ---| pi*I*|--- - ---| I*\/ 2 *|---- + log\\/ 2 /| ___ / ___\ I*\/ 2 *|- ---- + log\\/ 2 /| ___ / ___\
/ 1 I \ /pi*I / ___\\ / 1 I \ / pi*I / ___\\ / 1 I \ /3*pi*I / ___\\ / 1 I \ / 3*pi*I / ___\\ / 1 I \ / 1 I \ \/ 2 *\pi*I + log\-1 + \/ 2 // \/ 2 *log\\/ 2 / \/ 2 *\pi*I + log\\/ 2 // \/ 2 *log\1 + \/ 2 / \128 128/ \128 128/ \ 2 / I*\/ 2 *log\1 - I*\/ 2 / \ 2 / I*\/ 2 *log\1 + I*\/ 2 /
|- --- - ---|*|---- + log\\/ 2 /| + |- --- + ---|*|- ---- + log\\/ 2 /| + |--- - ---|*|------ + log\\/ 2 /| + |--- + ---|*|- ------ + log\\/ 2 /| - |- --- - ---|*log(2 + I) - |- --- + ---|*log(2 - I) - ------------------------------ - ---------------- + ------------------------- + -------------------- + ---------------- - ---------------- - --------------------------- - ------------------------ + ----------------------------- + ------------------------
\ 128 128/ \ 4 / \ 128 128/ \ 4 / \128 128/ \ 4 / \128 128/ \ 4 / \ 128 128/ \ 128 128/ 128 128 128 128 2 2 128 128 128 128
$$- \frac{\sqrt{2} \log{\left(\sqrt{2} \right)}}{128} + \frac{\sqrt{2} \log{\left(1 + \sqrt{2} \right)}}{128} - \frac{\sqrt{2} \left(\log{\left(-1 + \sqrt{2} \right)} + i \pi\right)}{128} + \left(\frac{1}{128} + \frac{i}{128}\right) \left(\log{\left(\sqrt{2} \right)} - \frac{3 i \pi}{4}\right) - \frac{i \pi \left(\frac{1}{128} - \frac{i}{128}\right)}{2} - \left(- \frac{1}{128} + \frac{i}{128}\right) \log{\left(2 - i \right)} + \left(- \frac{1}{128} - \frac{i}{128}\right) \left(\log{\left(\sqrt{2} \right)} + \frac{i \pi}{4}\right) - \frac{\sqrt{2} i \log{\left(1 - \sqrt{2} i \right)}}{128} - \frac{\sqrt{2} i \left(\log{\left(\sqrt{2} \right)} + \frac{i \pi}{2}\right)}{128} + \frac{\sqrt{2} i \left(\log{\left(\sqrt{2} \right)} - \frac{i \pi}{2}\right)}{128} + \frac{\sqrt{2} i \log{\left(1 + \sqrt{2} i \right)}}{128} + \left(- \frac{1}{128} + \frac{i}{128}\right) \left(\log{\left(\sqrt{2} \right)} - \frac{i \pi}{4}\right) - \left(- \frac{1}{128} - \frac{i}{128}\right) \log{\left(2 + i \right)} + \frac{i \pi \left(\frac{1}{128} + \frac{i}{128}\right)}{2} + \left(\frac{1}{128} - \frac{i}{128}\right) \left(\log{\left(\sqrt{2} \right)} + \frac{3 i \pi}{4}\right) + \frac{\sqrt{2} \left(\log{\left(\sqrt{2} \right)} + i \pi\right)}{128}$$
(-1/128 - i/128)*(pi*i/4 + log(sqrt(2))) + (-1/128 + i/128)*(-pi*i/4 + log(sqrt(2))) + (1/128 - i/128)*(3*pi*i/4 + log(sqrt(2))) + (1/128 + i/128)*(-3*pi*i/4 + log(sqrt(2))) - (-1/128 - i/128)*log(2 + i) - (-1/128 + i/128)*log(2 - i) - sqrt(2)*(pi*i + log(-1 + sqrt(2)))/128 - sqrt(2)*log(sqrt(2))/128 + sqrt(2)*(pi*i + log(sqrt(2)))/128 + sqrt(2)*log(1 + sqrt(2))/128 + pi*i*(1/128 + i/128)/2 - pi*i*(1/128 - i/128)/2 - i*sqrt(2)*(pi*i/2 + log(sqrt(2)))/128 - i*sqrt(2)*log(1 - i*sqrt(2))/128 + i*sqrt(2)*(-pi*i/2 + log(sqrt(2)))/128 + i*sqrt(2)*log(1 + i*sqrt(2))/128
Use the examples entering the upper and lower limits of integration.