Trig substitution is the way to go here. Do you know the substitution to use?cjarnutowski said:Homework Statement
∫1/(x2+4)2
Homework Equations
∫dx/(x2+a2)=(1/a)arctan(x/a)+c
The Attempt at a Solution
This looks like an arctan integral or a trig substitution, but in its current form neither would work without manipulation. I'm mainly looking for how to manipulate it to use one of these methods
Thanks!
Arctan and trig-sub methods are typically used when the integrand (the expression being integrated) contains a trigonometric function, such as sine, cosine, tangent, or secant. These methods can also be used when the integrand contains a square root or a quadratic expression.
Yes, it is possible to use both arctan and trig-sub methods in the same integral. In fact, sometimes using both methods can make the integral easier to solve.
To use arctan, you will need to rewrite the integrand in terms of the derivative of arctan (1/(1+x^2)) and then use integration by parts to simplify the integral. This method is particularly useful when the integrand contains a fraction or a product of functions.
Trig-sub, short for trigonometric substitution, involves replacing a variable in the integral with a trigonometric function. This allows us to use trigonometric identities and formulas to simplify the integral. The key is to choose the appropriate substitution based on the form of the integrand.
No, these methods may not work for all integrals. It is important to know other integration techniques, such as u-substitution and partial fractions, in order to solve integrals that cannot be manipulated using trig-sub or arctan.