Integrate the function: \(\frac {x-1}{\sqrt {x^2-1}}\)
Solution and Explanation
\(∫\frac {x-1}{\sqrt {x^2-1}}\ dx\) = \(∫\frac {x}{\sqrt {x^2-1}}\ dx\) - \(∫\frac {1}{\sqrt {x^2-1}}\ dx\) ……....(1)
\(For ∫\frac {x}{\sqrt {x^2-1}} dx, \ let x^2-1 = t ⇒ 2x dx = dt\)
\(∴ ∫\frac {x}{\sqrt {x^2-1}} dx = \frac 12 ∫\frac {dt}{\sqrt t}\)
\(=\frac 12 ∫t^{-\frac 12} dt\)
\(=\frac 12[2t^{\frac 12}]\)
\(=\sqrt t\)
\(=\sqrt {x^2-1}\)
\(From\ (1), we\ obtain\)
\(∫\frac {x-1}{\sqrt {x^2-1}}\ dx\) = \(∫\frac {x}{\sqrt {x^2-1}}\ dx\) - \(∫\frac {1}{\sqrt {x^2-1}}\ dx\) \([∫\frac {1}{\sqrt {x^2-a^2}} \ dt = log\ |x+\sqrt {x^2-a^2|}]\)
\(= \sqrt {x^2-1}-log|x+\sqrt {x^2-1}|+C\)
Top Questions on integral
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Questions Asked in CBSE CLASS XII exam
- Find the inverse of each of the matrices, if it exists. \(\begin{bmatrix} 1 & 3\\ 2 & 7\end{bmatrix}\)
- For what values of x,\(\begin{bmatrix} 1 & 2 & 1 \end{bmatrix}\)\(\begin{bmatrix} 1 & 2 & 0\\ 2 & 0 & 1 \\1&0&2 \end{bmatrix}\)\(\begin{bmatrix} 0 \\2\\x\end{bmatrix}\)=O?
- Find the inverse of each of the matrices,if it exists \(\begin{bmatrix} 2 & 1 \\ 7 & 4 \end{bmatrix}\)
What is the Planning Process?
- CBSE CLASS XII - 2023
- Planning process steps
- Find the inverse of each of the matrices,if it exists. \(\begin{bmatrix} 2 & 3\\ 5 & 7 \end{bmatrix}\)
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.
