- #1
Albert1
- 1,221
- 0
\[\int_{0}^{1}\dfrac{x^4(1-x)^4}{1+x^2}dx=?\]
Last edited:
Albert said:\[\int_{0}^{1}\dfrac{x^4(1-x)^4}{1+x^2}dx=?\]
kaliprasad said:expanding we get
$x^6-4x^5+5x^4-4x^2+4-\frac{4}{1+x^2}$
integrating we get
$\frac{x^7}{7} – \frac{2}{3}x^6+ x^5-\frac{4}{3}x^2 + 4x – 4\arctan x$
x=1 gives $\frac{1}{7} – \frac{2}{3}+1-\frac{4}{3} + 4 – \pi = \frac{22}{7}- \pi$ and at x =0 it is zero hence the integral is $\frac{22}{7}- \pi$
Albert said:answer correct , a typo exists in your intergrating
An integral is a mathematical concept that represents the area under a curve in a graph. It is a fundamental tool in calculus and is used to solve a variety of problems in mathematics and science.
The process for solving an integral involves finding the antiderivative of the given function and then evaluating it at the upper and lower limits of integration. In other words, you need to find a function whose derivative is equal to the given function and then plug in the upper and lower limits to find the area under the curve.
To solve this integral, you can use techniques such as substitution, integration by parts, or partial fraction decomposition. After applying one of these methods, you can then use algebraic manipulation to simplify the integrand and evaluate the integral at the limits of integration.
The limits of integration represent the boundaries of the area under the curve that we are trying to find. They determine the starting and ending points of the integration process and are crucial in finding the exact value of the integral.
Solving integrals is important in science because many natural phenomena can be modeled by mathematical functions, and finding the area under these functions can provide valuable information about these phenomena. Integrals are also used in many scientific equations and formulas, making them essential in various fields such as physics, engineering, and economics.