Evaluate the following integrals or show that the integral diverges. ∫−∞∞x31+x8dx

Question

Evaluate the following integrals or show that the integral diverges.  ∫−∞∞x31+x8dx

nghodlokl · Accepted Answer

Step 1 Given- ∫−∞∞x31+x8dx To evaluate- The above improper integral. Concept Used- The above integral can be solved by using the substitution method. Step 2 Explanation- Rewrite the given integral as, I=∫−∞∞x31+x8dx =∫−∞∞x31+(x4)2dx Now, substituting x4=t and differentiating w.r.t. x, we get, 4x3dx=dt x3dx=dt4 At x=−∞,t=−∞ and at x=∞,t=∞. Now, from the above integral,we can write as, Step 3 =∞−∞∞dt41+t2 =14∫−∞∞11+t2dt =14[tan−1(t)]−∞+∞ =14[tan−1(∞)−tan−1(−∞)] =14[π2−(−π2)] =π4 Answer- Hence, the value of the integral ∫−∞∞x31+x8dx is π4.

Marcus Herman · Accepted Answer

It is required to calculate:  ∫x3x8+1dx  =14∫1u2+1du  Now we calculate:  ∫1u2+1du  This is the well-known tabular integral:  =arctan⁡(u)  We substitute the already calculated integrals:  14∫1u2+1du  =arctan⁡(u)4  Reverse replacement u=x4:  =arctan⁡(x4)4  ∫x3x8+1dx  =arctan⁡(x4)4+C

nick1337 · Accepted Answer

We put the expression 4∗x3 under the differential sign, i.e.: 4x3dx=d(x4),t=x4 Then the original integral can be written as follows: ∫14∗(t2+1)∗dt This is a table integral: ∫14∗(t2+1)∗dt=arctan⁡(t)4+C To write down the final answer, it remains replace t with x4 arctan⁡(x4)4+C

Evaluate the following integrals or show that the integral diverges. \int_{-\infty}^{\infty}\frac{x^{3}}{1+x^{8}}dx

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