For the following integral find an appropriate TRIGONOMETRIC SUBSTITUTION of the form x=f(t) to simplify the integral.
INT (x^2)/(sqrt(7x^2+4))dx dx x=?
This is the first of several YouTube clips where Khan shows in nice detail how to work trig substitution.
http://www.youtube.com/watch?v=n4EK92CSuBE&feature=youtube_gdata
This is part of
http://www.khanacademy.org/
which is a wonderful collection of hundreds of YouTube clips showing most of the mathematical procedures up to college level Calculus.
Let me know if this was helpful
neither of these videos explain the situation where vaiable x is in the numerator, so i still cant solve it
To find an appropriate trigonometric substitution, we can start by letting \(x = f(t)\). We want to choose a substitution that will simplify the integral by getting rid of the square root term. In this case, it is helpful to notice that the expression inside the square root, \(7x^2 + 4\), resembles a perfect square, specifically, \(a^2 + b^2\).
To transform the integrand accordingly, we can rewrite \(7x^2 + 4\) as \((\sqrt{7}x)^2 + 2^2\). We can now make the substitution \(x = \frac{2}{\sqrt{7}}\tan(t)\), which assumes the form \(x=f(t)\).
To find \(dx\) in terms of \(dt\) we differentiate the substitution \(x = \frac{2}{\sqrt{7}}\tan(t)\) with respect to \(t\):
\(\frac{dx}{dt} = \frac{2}{\sqrt{7}}\sec^2(t)\)
Rearranging, we have:
\(dx = \frac{2}{\sqrt{7}}\sec^2(t) dt\)
Now, we can rewrite the integral using the trigonometric substitution:
\(\int \frac{x^2}{\sqrt{7x^2+4}} dx\) becomes \(\int \frac{(\frac{2}{\sqrt{7}}\tan(t))^2}{\sqrt{7(\frac{2}{\sqrt{7}}\tan(t))^2 + 4}} \cdot \frac{2}{\sqrt{7}}\sec^2(t) dt\)
Simplifying this expression, we get:
\(\int \frac{4\tan^2(t)}{\sqrt{\frac{28}{7}\tan^2(t) + 4}} \cdot \frac{2}{\sqrt{7}}\sec^2(t) dt\)
Further simplification yields:
\(\int \frac{8\tan^2(t)}{\sqrt{28\tan^2(t) + 28}} dt\)
We can now simplify the integral using the trigonometric identity \(\tan^2(t) + 1 = \sec^2(t)\):
\(\int \frac{8(\sec^2(t) - 1)}{\sqrt{28\tan^2(t) + 28}} dt\)
After simplification, we arrive at:
\(\int \frac{8\sec^2(t) - 8}{\sqrt{28\tan^2(t) + 28}} dt\)
This integral can now be evaluated by applying appropriate trigonometric identities and integration techniques.