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No, u= sin(arctan(x))transgalactic said:but the "u" expression represents x
For something simple, like sin(tan-1(x)), you can imagine a triangle with "opposite side" x and "near side" 1 so that "opposite side over near side" = tan(angle)= x and angle= tan-1(x). Then sin(tan-1(x))= sin(angle) which is "near side over hypotenuse". Use the Pythagorean theorem to find the length of the hypotenus: [itex]\sqrt{x^2+ 1}[/itex] and you get sin(tan-1(x))= [itex]x/\sqrt{x^2+ 1}[/itex].i heard of some triangle method of solving stuff like
[itex]
\sin(2\tan^{-1}(x))
[/itex]
??
One tip is to try to recognize patterns or familiar forms within the integral. Another is to use substitution or integration by parts to break down the integral into smaller, more manageable parts.
The limits of integration can often be determined by looking at the original problem or by using substitution to solve for the new limits. In some cases, it may be necessary to graph the original function to determine the appropriate limits.
One common mistake is to forget to account for the change in limits when using substitution. Another is to confuse the order of integration when using integration by parts. It's also important to carefully check for algebraic errors when simplifying the integral.
You can check your answer by taking the derivative of the simplified integral and comparing it to the original function. You can also use a graphing calculator to plot both the original function and the simplified integral and see if they match.
Yes, there are some common integrals that have well-known solutions and can be used as shortcuts. It's also helpful to memorize some basic integration formulas, such as the power rule and trigonometric identities, to simplify the process.