\(\int \sqrt{x^2-8x+7}dx\) is equal to
\(\int \sqrt{x^2-8x+7}dx\) is equal to
\(\frac{1}{2}(x-4)\sqrt{x^2-8x+7}+9\log\mid x-4+\sqrt{x^2-8x+7}\mid+C\)
\(\frac{1}{2}(x+4)\sqrt{x^2-8x+7}+9\log\mid x+4+\sqrt{x^2-8x+7}\mid+C\)
\(\frac{1}{2}(x-4)\sqrt{x^2-8x+7}-3\sqrt 2\log\mid x-4+\sqrt{x^2-8x+7}\mid+C\)
\(\frac{1}{2}(x-4)\sqrt{x^2-8x+7}-\frac{9}{2}log\mid x-4+\sqrt{x^2-8x+7}\mid+C\)
The Correct Option is D
Solution and Explanation
Let \(I=\int\sqrt{x^2-8x+7}dx\)
=\(\int \sqrt{(x^2-8x+16)-9}dx\)
=\(\int\sqrt{(x-4)^2-(3)^2}dx\)
It is known that,\(\int\sqrt{x^2-a^2}dx=\frac{x}{2}\sqrt{x^2-a^2}-\frac{a^2}{2}\log\mid x+\sqrt{x^2-a^2}\mid+C\)
∴\(I=\frac{(x-4)}{2}\sqrt{x^2-8x+7}-\frac{9}{2}\log\mid (x-4)+\sqrt{x^2-8x+7}\mid+C\)
Hence, the correct answer is D.
Top Questions on integral
- Let $f : (0, \infty) \to \mathbb{R}$ and $F(x) = \int_{0}^{x} tf(t) \, dt$. If $F(x^2) = x^4 + x^5$, then \[\sum_{r=1}^{12} f(r^2)\]is equal to:
- If \[\int_{0}^{\frac{\pi}{3}} \cos^4 x \, dx = a\pi + b\sqrt{3},\]where $a$ and $b$ are rational numbers, then $9a + 8b$ is equal to:
- The value of \[\int_{0}^{1} \left(2x^3 - 3x^2 - x + 1\right)^{\frac{1}{3}} \, dx\]is equal to:
- Let \( r_k = \frac{\int_{0}^{1} (1 - x^7)^k \, dx}{\int_{0}^{1} (1 - x^7)^{k+1} \, dx}, \, k \in \mathbb{N} \). Then the value of \[ \sum_{k=1}^{10} \frac{1}{7(r_k - 1)}\]is equal to ______.
- \( \int_{0}^{\pi/4} \frac{\cos^2 x \sin^2 x}{\left( \cos^3 x + \sin^3 x \right)^2} \, dx \) is equal to:
Questions Asked in CBSE CLASS XII exam
- Find the inverse of each of the matrices,if it exists \(\begin{bmatrix} 2 & 1 \\ 7 & 4 \end{bmatrix}\)
- 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?
What is the Planning Process?
- CBSE CLASS XII - 2023
- Planning process steps
- Find the inverse of each of the matrices,if it exists. \(\begin{bmatrix} 2 & 3\\ 5 & 7 \end{bmatrix}\)
- Find the inverse of each of the matrices, if it exists. \(\begin{bmatrix} 1 & 3\\ 2 & 7\end{bmatrix}\)
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.
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