Integrate the function: \(\frac {x^2}{\sqrt {x^6+a^6}}\)
Solution and Explanation
\(Let \ x^3 = t\)
\(⇒ 3x^2 dx = dt\)
\(∴ ∫\frac {x^2}{\sqrt {x^6+a^6}} dx = \frac 13 ∫\frac {dt}{\sqrt {t^2+(a^3)^2}}\)
\(=\frac 13\ log\ |t+\sqrt {t^2+a^6}|+C\)
\(=\frac 13\ log\ |x^3+\sqrt {x^6+a^6}|+C\)
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Concepts Used:
Integrals of Some Particular Functions
There are many important integration formulas which are applied to integrate many other standard integrals. In this article, we will take a look at the integrals of these particular functions and see how they are used in several other standard integrals.
Integrals of Some Particular Functions:
- ∫1/(x2 – a2) dx = (1/2a) log|(x – a)/(x + a)| + C
- ∫1/(a2 – x2) dx = (1/2a) log|(a + x)/(a – x)| + C
- ∫1/(x2 + a2) dx = (1/a) tan-1(x/a) + C
- ∫1/√(x2 – a2) dx = log|x + √(x2 – a2)| + C
- ∫1/√(a2 – x2) dx = sin-1(x/a) + C
- ∫1/√(x2 + a2) dx = log|x + √(x2 + a2)| + C
These are tabulated below along with the meaning of each part.
