applestrudle said:Homework Statement
Integrate sqrt(1+4x^2)
limits x = 0 to x = 1
Homework Equations
The Attempt at a Solution
let 2x = sinhu
I = 0.5 ∫ (coshu)^2du
limits u = 0 u = sinh^-1(2)
0.5 ∫ (e^2u +e^-2u +2)/4 du
0.5 [(e^2u)/8 -(e^-2u)/8 +2u]
I = 2.56 which is wrong :(
why is it wrong?
please help!
The basic concept of integration is finding the area under a curve. It is the inverse process of differentiation and involves finding the antiderivative of a function.
To integrate a square root function, you can use the substitution method or integration by parts. In this case, we can use the substitution method by letting u = 1 + 4x^2, which simplifies the integral to ∫√u du.
The general formula for integrating √(1+4x^2) is ∫√(1+4x^2) dx = (1/4)∫√u du = (1/8)(u√u + ln|u|) + C, where u = 1 + 4x^2.
No, the integral of √(1+4x^2) cannot be evaluated using basic integration rules. It requires the use of advanced integration techniques such as substitution or integration by parts.
The integration of √(1+4x^2) is commonly used in physics and engineering to calculate the work done by a varying force. It is also used in calculating the arc length of a curve in mathematics and has applications in other fields such as economics and biology.