Question

How will you integrate, the given expression \[\int {\dfrac{{dx}}{{{{(1 + {x^4})}^{\dfrac{1}{4}}}}}} \] ?

Answer 1

Hint: Here we have to integrate the given question, to integrate the following expression we first need to transfer the denominator term into numerator and then a minus sign will appear with the power of the expression, and then we need to do normal integration to get the solution.

Formulae Used:
Formulae for integration of similar terms is,
\[\int {{{(1 + {x^n})}^{ - y}}} dx = \dfrac{1}{{ - y + 1}}{(1 + {x^n})^{ - y + 1}}(x + \dfrac{1}{{n + 1}}{x^{n + 1}})\]

Complete step by step solution:
The given question is \[\int {\dfrac{{dx}}{{{{(1 + {x^4})}^{\dfrac{1}{4}}}}}} \].First we have to transfer the denominator part into numerator and then solve further, on solving we get:
\[\int {\dfrac{{dx}}{{{{(1 + {x^4})}^{\dfrac{1}{4}}}}}} = \int {{{(1 + {x^4})}^{ - \dfrac{1}{4}}}} dx\]
Here the question now becomes like the above expression now we have to use the integration as:
\[\int {{{(1 + {x^n})}^{ - y}}} dx = \dfrac{1}{{ - y + 1}}{(1 + {x^n})^{ - y + 1}}(x + \dfrac{1}{{n + 1}}{x^{n + 1}})\]
Now applying this to our expression we get:
\[\int {{{(1 + {x^4})}^{ - \dfrac{1}{4}}}} dx = \dfrac{1}{{ - \dfrac{1}{4} + 1}}{(1 + {x^4})^{ - \dfrac{1}{4} + 1}}(x + \dfrac{1}{{4 + 1}}{x^{4 + 1}}) \\
\therefore \int {{{(1 + {x^4})}^{ - \dfrac{1}{4}}}} dx = \dfrac{4}{3}{(1 + {x^4})^3}(x + \dfrac{1}{5}{x^5})\]
Hence we get the final result of integration of the required expression asked in the question, here we can cross check the answer by solving in last to top mode by differentiating the result till we get the expression given in the question, or here we can also put the value of “x” and then check for the left hand side value equal to right hand side.

Note: Here the given question can also be solved by using By-parts or by using any other method of integration, integration is something which can be done in a lot of ways, and the solution may or may not match every time, but if the steps used are correct then all the expressions obtained will be correct. Here to check the correctness one can put any assumed value and check for the result.