Integrate the function: \(\sqrt{1-4x-x^2}\)
Solution and Explanation
Let \(I=\int \sqrt{1-4x-x^2}dx\)
=\(\int \sqrt{1-(x^2+4x+4-4)}dx\)
=\(\int \sqrt{1+4-(x+2)^2}dx\)
=\(\int\sqrt{(\sqrt5)^2-(x+2)^2}dx\)
It is known that,\(\int \sqrt{a^2-x^2}dx=\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}\sin^{-1}\frac{x}{a}+C\)
∴\(I=\frac{(x+2)}{2}\sqrt{1-4x-x^2}+\frac{5}{2}\sin^{-1}\bigg(\frac{x+2}{\sqrt 5}\bigg)+C\)
Top Questions on integral
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Questions Asked in CBSE CLASS XII exam
Standard electrode potential for \( \text{Sn}^{4+}/\text{Sn}^{2+} \) couple is +0.15 V and that for the \( \text{Cr}^{3+}/\text{Cr} \) couple is -0.74 V. The two couples in their standard states are connected to make a cell. The cell potential will be:
To calculate the cell potential (\( E^\circ_{\text{cell}} \)), we use the standard electrode potentials of the given redox couples.
Given data:
\( E^\circ_{\text{Sn}^{4+}/\text{Sn}^{2+}} = +0.15V \)
\( E^\circ_{\text{Cr}^{3+}/\text{Cr}} = -0.74V \)- CBSE CLASS XII - 2025
- Electrochemistry
- “You speak peculiar things.” Why does Derry find Mr. Lamb’s conversation peculiar? (On the Face of It)
मोबाइल फोन विहीन दुनिया — 120 शब्दों में रचनात्मक लेख लिखिए :
- CBSE CLASS XII - 2025
- प्रश्न उत्तर
- Emily, Flora and Ginni entered into a partnership on 1st October, 2023 with capitals of ₹ 10,00,000 each. The partnership deed provided for interest on capital at 10% p.a. The firm earned a net profit of ₹ 7,50,000 for the year ended 31st March, 2024. The amount of profit transferred to Emily’s capital account was :
- CBSE CLASS XII - 2025
- Partnership Accounts
- "The Government had launched Incredible India Campaign to promote tourism in various parts of the country."
Elaborate the impact of Incredible India Campaign on foreign exchange reserves and Balance of Payment of India.- CBSE CLASS XII - 2025
- Balance of payments (BOP)
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.
