Integrate the function: \(\sqrt{1+3x-x^2}\)
Solution and Explanation
Let \(I= \int \sqrt{1+3x-x^2}dx\)
=\(\int \sqrt{1-\bigg(x^2-3x+\frac{9}{4}-\frac{9}{4}\bigg)}dx\)
=\(\int\sqrt{\bigg(1+\frac{9}{4}\bigg)-\bigg(x-\frac{3}{2}\bigg)^2}dx\)
=\(\int\sqrt{\bigg(\frac{\sqrt13}{2}\bigg)^2-\bigg(x-\frac{3}{2}\bigg)^2}dx\)
It is known that,\(\int\sqrt{a^2-x^2}dx=\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}\sin^{-1}\frac{x}{a}+C\)
∴\(I= \frac{x-\frac{3}{2}}{2}\sqrt{1+3x-x^2}+\frac{13}{4*2}\sin^{-1}\bigg(\frac{x-\frac{3}{2}}{\frac{\sqrt 13}{2}}\bigg)+C\)
=\(\frac{2x-3}{4}\sqrt{1+3x-x^2}+\frac{13}{8}\sin^{-1}\bigg(\frac{2x-3}{\sqrt 13}\bigg)+C\)
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R = {(a, b): a ≤ b2 } is neither reflexive nor symmetric nor transitive.Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as
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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.
