Integrate the function: \(\sqrt{x^2+3x}\)
Solution and Explanation
Let\(I= \int \sqrt{x^2+3x} \: dx\)
=\(\int \sqrt{x^2+3x+\frac{9}{4}-\frac{9}{4}}dx\)
=\(\int \sqrt{\bigg(x+\frac{3}{2}\bigg)^2-\bigg(\frac{3}{2}\bigg)^2}dx\)
It is known that,\(\int \sqrt{x^2-a^2}dx=\frac{x}{2}\sqrt{x^2-a^2}-\frac{a^2}{2}\log\mid x+\sqrt{x^2-a^2\mid}+C\)
∴\(I= \frac{\bigg(x+\frac{3}{2}\bigg)}{2}\sqrt{x^2+3x}-\frac{\frac{9}{4}}{2}\log \mid \bigg(x+\frac{3}{2}\bigg)+\sqrt{x^2+3x}\mid+C\)
=\(\frac{(2x+3)}{4}\sqrt{x^2+3x}-\frac{9}{8}\log \mid \bigg(x+\frac{3}{2}\bigg)+\sqrt{x^2+3x}\mid+C\)
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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.
