\(\int \sqrt{1+x^2}dx\) is equal to
\(\int \sqrt{1+x^2}dx\) is equal to
\(\frac{x}{2}\sqrt{1+x^2}+\frac{1}{2}\log\mid x+\sqrt{1+x^2}\mid +C\)
\(\frac{2}{3}(1+x^2)^{\frac{2}{3}}+C\)
\(\frac{2}{3}x(1+x^2)^{\frac{3}{2}}+C\)
\(\frac{x^2}{2}\sqrt{1+x^2}+\frac{1}{2}x^2\log\mid x+\sqrt{1+x^2}\mid+C\)
The Correct Option is A
Solution and Explanation
It is known that,\(\int \sqrt{a^2+x^2}dx = \frac{x}{2}\sqrt{a^2+x^2}+\frac{a^2}{2}\log\mid x+\sqrt{x^2+a^2}\mid+C\)
∴\(\int \sqrt{1+x^2}dx=\frac{x}{2}\sqrt{1+x^2}+\frac{1}{2}\log\mid x+\sqrt{1+x^2}\mid+C\)
Hence, the correct answer is A.
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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.
