Example Question - integration techniques
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Techniques for Solving Trigonometric Integrals
The integral that you've shown in the image is: \[ \int_{0}^{\frac{\pi}{2}} \frac{\cos x}{(4 + \sin x)^2} dx \] This integral can be solved using a substitution that exploits the symmetry of the integrand over the interval \([0, \frac{\pi}{2}]\), plus some trigonometric identities. We will start by using the substitution \(u = \frac{\pi}{2} - x\), which implies \(\sin x = \sin(\frac{\pi}{2} - u) = \cos u\) and \(\cos x = \cos(\frac{\pi}{2} - u) = \sin u\). The differential \(dx\) becomes \(-du\). The limits of integration also change with this substitution. When \(x = 0\), \(u = \frac{\pi}{2}\), and when \(x = \frac{\pi}{2}\), \(u = 0\). The integral becomes: \[ \int_{\frac{\pi}{2}}^{0} \frac{\sin u}{(4 + \cos u)^2} (-du) \] Since the limits are now reversed, we can change them back and remove the negative sign: \[ \int_{0}^{\frac{\pi}{2}} \frac{\sin u}{(4 + \cos u)^2} du \] We can now add the two integrals together since they are equivalent: \[ \frac{1}{2}\int_{0}^{\frac{\pi}{2}} \left( \frac{\cos x}{(4 + \sin x)^2} + \frac{\sin x}{(4 + \cos x)^2} \right) dx \] Notice something special here: the integrand has a symmetry that allows us to simplify it using the Pythagorean trigonometric identity (\(\sin^2 x + \cos^2 x = 1\)) or by using some partial fraction expansion. However, we can notice that both terms have the same denominator on the interval of integration when the identities \(\cos(\frac{\pi}{2}-x) = \sin x\) and \(\sin(\frac{\pi}{2}-x) = \cos x\) are considered. Using these identities and combining the two integrals, we find that the resulting integrand is an even function over the symmetric interval \([- \frac{\pi}{2}, \frac{\pi}{2}]\). We can exploit this symmetry to compute the integral over just half the interval and multiply the result by 2: \[ 2 \times \frac{1}{2}\int_{0}^{\frac{\pi}{2}} \frac{d(x - \frac{\pi}{2})}{(4 + \sin x)^2} \] The derivative \(d(x - \frac{\pi}{2})\) on the numerator is essentially a differential form that's equivalent to just \(dx\), so we don't need to worry about it affecting the value of the integral. Finally, we can integrate: \[ \int_{0}^{\frac{\pi}{2}} \frac{dx}{(4 + \sin x)^2} \] At this stage, the integral might be more straightforwardly computed by realizing that a standard trigonometric substitution, integrating by parts, or using a trigonometric identity could resolve the integrand. However, without a more specific method or simplification, the integral may become quite cumbersome to evaluate by hand. For more complex trigonometric integrals like the one in question, oftentimes numerical integration techniques or the assistance of computational tools are required to find the precise value of the integral.
Definite Integrals of Polynomial Functions
The image shows an integral which needs to be solved. The function to be integrated is a polynomial: 8x^3 - x^2 + 5x - 1. To solve the integral, we need to integrate each term individually with respect to x. Here's how you integrate each term: 1. ∫8x^3 dx = (8/4)x^4 = 2x^4 2. ∫(-x^2) dx = -(1/3)x^3 3. ∫5x dx = (5/2)x^2 4. ∫(-1) dx = -x Now, putting it all together: ∫(8x^3 - x^2 + 5x - 1) dx = 2x^4 - (1/3)x^3 + (5/2)x^2 - x + C Where C represents the constant of integration.
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