find the integral of x/(x^2+4)dx using trigonometric substitution.
I don't understand why you were instructed to use trig substitution for this question, it is straightforward
You should recognize certain pattern of derivatives and integrals
Notice that the derivative of the denominator is 2x and we have x at the top, so this follows the pattern of log derivatives directly
If we had ∫2x/(x^2 + 4) dx it would simply be ln(x^2 + 4) + c
so for
∫ x/(x^2+4) dx we would get (1/2) ln(x^2 + 4) + c
we are supossed to do it both ways. i got it using the u-substitution method. we are supossed to show that the answers are equivalent.
To find the integral of x/(x^2+4)dx using trigonometric substitution, follow these steps:
Step 1: Identify the appropriate trigonometric substitution.
In this case, x^2 + 4 resembles the form of a^2 + u^2, which can be rewritten in terms of trigonometric functions. Let u = 2tan(theta), where theta is an angle.
Step 2: Calculate dx in terms of d(theta).
Differentiate both sides of the equation u = 2tan(theta) with respect to theta, and then solve for dx. Taking the derivative of u with respect to theta gives du/d(theta), and differentiating tan(theta) gives sec^2(theta), so we have du = 2sec^2(theta) d(theta). Dividing both sides by 2sec^2(theta) gives du/(2sec^2(theta)) = d(theta). Since sec^2(theta) = 1 + tan^2(theta), we can simplify the equation to du/(2(1+tan^2(theta))) = d(theta). Rearranging, we get du/(2sec^2(theta)) = d(theta), or du/(2u^2 + 4) = d(theta).
Step 3: Substitute x and dx in terms of theta.
Using the substitution u = 2tan(theta), we can determine the value of x in terms of theta. Since u = 2tan(theta), tan(theta) = u/2, and by taking the inverse tangent on both sides, we get theta = atan(u/2). To find dx in terms of theta, we substitute x = 2tan(theta) into dx = du/(2u^2 + 4). This gives dx = (2sec^2(theta)) d(theta).
Step 4: Rewrite the integral using trigonometric substitution.
Substituting x and dx in terms of theta, we can rewrite the integral as ∫((2tan(theta))(2sec^2(theta)))/(4tan^2(theta) + 4) d(theta).
Step 5: Simplify the integral.
Simplifying the expression in the integral gives ∫4sec^2(theta)tan(theta) / (4(tan^2(theta) + 1)) d(theta). Canceling out the 4 and rearranging the terms, we have ∫sec^2(theta)tan(theta) / (tan^2(theta) + 1) d(theta).
Step 6: Use a trigonometric identity to simplify further.
Using the identity 1 + tan^2(theta) = sec^2(theta), we can rewrite the integral as ∫tan(theta) d(theta), which can be easily integrated.
Step 7: Integrate the expression.
The integral of tan(theta) with respect to theta is -ln|cos(theta)| + C, where C is the constant of integration.
Step 8: Substitute back in terms of x.
We substitute theta = atan(u/2) back into the expression, giving the final answer of -ln|cos(atan(u/2))| + C.
Step 9: Simplify the expression further.
Using the identity cos(atan(u/2)) = 2 / √(u^2 + 4), we can simplify the expression to -ln|2 / √(u^2 + 4)| + C.
Step 10: Substitute x back in terms of u.
Recall that u = 2tan(theta). From here, we can substitute back x = 2tan(theta) to obtain the final answer:
-ln|2 / √(x^2 + 4)| + C, where C is the constant of integration.