Integration of UV Formula
Integration of uv formula is a convenient means of finding the integration of the product of the two functions u and v respectively. This formula helps in finding the integration of the functions in simple quick steps. Further, the functions used in this integration of uv formula can be algebraic expressions, trigonometric ratios, or logarithmic functions.
What is Integration of UV Formula?
The functions u and v are multiplied the integration is performed. For function u a derivative is found, and the function v is integrated twice across the formula, to get the final result. Integration of uv formula is given as:
\[ \int u.v = u.\int v.dx - \int(\int v.dx.u').dx \]
Let us try out a few examples to better understand how to apply the integration of uv formula.
Solved Examples on Integration of UV Formula
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Example 1: Find the integral of x.Sinx.
Solution:
\(\begin{align} \int x.Sinx.dx &= x \int Sinx.dx - \int(\int Sinx.dx. \frac{d}{dx}.x).dx \\&= -xCosx - \int(-Cosx . 1).dx \\&= -xCosx + \int Cosx.dx \\&=-xCosx + Sinx \\&=Sinx - xCosx + C\end{align} \)
Answer: \(\int x.sinx.dx = Sinx - xCosx + C \) -
Example 2: Find the integral of x2.logx
Solution:
\(\begin{align} \int x^2.logx .dx &= logx.\int x^2.dx - \int(\int x^2.dx.\dfrac{d}{dx}.logx).dx \\&= logx.\dfrac{x^3}{3} - \int(\dfrac{x^3}{3} .\dfrac{1}{x}).dx \\&=\dfrac{x^3}{3}.logx - \dfrac{1}{3} \int x^2 .dx \\&=\dfrac{x^3}{3}.logx - \dfrac{1}{3}.\dfrac{x^2}{3} + C \\&= \dfrac{x^3}{3}.logx - \dfrac{x^2}{9} + C\end{align} \)
Answer: \(\int x^2.logx = \dfrac{x^3}{3}.logx - \dfrac{x^2}{9} + C\)