Integrate the function: \(\frac {3x^2}{x^6+1}\)
Solution and Explanation
\(Let\ x^3 = t\)
\(∴ 3x^2 dx = dt\)
\(⇒ ∫\frac {3x^2}{x^6+1}dx = ∫\frac {dt}{t^2+1}\)
\(=tan^{-1} t+C\)
\(=tan^{-1} (x^3)+C\)
Top Questions on integral
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Questions Asked in CBSE CLASS XII exam
आदिकविः कः कथ्यते?
- CBSE CLASS XII - 2026
- संस्कृत साहित्य
'गन्तुम्' इति पदे कः प्रत्ययः प्रयुक्तः?
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'प्रतिदिनम्' इति पदस्य समासविग्रहं लिखत।
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अधोलिखितपदेषु विसर्गसन्धिं कुरुत - 'रामः + अपि'।
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कर्मगौरवम् इति पाठाधारेण कस्य अधिकारः कर्मणि एव अस्ति?
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.
