# Integration by substitution

• Nov 3rd 2010, 02:43 AM
Hellbent
Integration by substitution
Hi,

Is this correct? Else, may I have some assistance please?

If $x = 4\cos^2 \theta + 7 \sin^2\theta$, show that $7 - x = 3\cos^2\theta$, and find a similar expression for $x - 4$. By using substitution $x = 4cos^2\theta + 7\sin^2\theta$, evaluate $\int_4^7 \frac{1}{\sqrt{x - 4)}(7 - x)}\,dx$

$x = 4\cos^2\theta + 7\sin^2\theta$

$x = 4\cos^2\theta + 7(1 - cos^2\theta)$

$( x = 4\cos^2\theta + 7 - 7cos^2\theta) \times -1$

$- x = - 4\cos^2\theta - 7 + 7cos^2\theta$

$7 - x = 3 \cos^2\theta$

$x = 4\cos^2\theta + 7 \sin^2\theta$

$x = 4(1 - \sin^2\theta) + 7\sin^2\theta$

$x = 4 - 4\sin^2\theta + 7 \sin^2\theta$

$x - 4 = 3 \sin^2\theta$

$\int_4^7 \frac{1}{\sqrt{(x - 4)(7 - x)}}d\theta$

$\int_4^7 \frac{1}{\sqrt{(3\sin^2\theta)(3\cos^2\theta)}}d\t heta$

$\int_0^{90} \frac{6\cos\theta\sin\theta}{\sqrt{(3\sin^2\theta) (3\cos^2\theta)}}d\theta$

$\int_0^{90} \frac{3\sin 2\theta}{\sqrt{9\cos^2\theta - 9\cos^4\theta}}$

$\int_0^{90} \frac{3\sin2\theta}{3\sqrt{cos^2\theta - \cos^4\theta}}d\theta$

$\int_0^{90} \frac{\sin 2\theta}{\sqrt{\cos^2\theta(1 - cos^2\theta)}}d\theta$

$\int_0^{90} \frac{2\sin\theta}{cos\theta\sqrt{(1 - cos^2\theta)}}d\theta$

$2 \sin\theta \int_0^{90} \frac{1}{1 - \cos^2\theta}d\theta$

$\big[2\sin\theta \cdot \sin^{-1}(\cos\theta\big]_0^{90}$

$[2\cdot 1 \cdot \sin^{-1}(0)] - [0 \cdot 0]$

$0$
• Nov 3rd 2010, 02:58 AM
tonio
Quote:

Originally Posted by Hellbent
Hi,

Is this correct? Else, may I have some assistance please?

If $x = 4\cos^2 \theta + 7 \sin^2\theta$, show that $7 - x = 3\cos^2\theta$, and find a similar expression for $x - 4$. By using substitution $x = 4cos^2\theta + 7\sin^2\theta$, evaluate $\int_4^7 \frac{1}{\sqrt{x - 4)}(7 - x)}\,dx$

$x = 4\cos^2\theta + 7\sin^2\theta$

$x = 4\cos^2\theta + 7(1 - cos^2\theta)$

$( x = 4\cos^2\theta + 7 - 7cos^2\theta) \times -1$

$- x = - 4\cos^2\theta - 7 + 7cos^2\theta$

$7 - x = 3 \cos^2\theta$

$x = 4\cos^2\theta + 7 \sin^2\theta$

$x = 4(1 - \sin^2\theta) + 7\sin^2\theta$

$x = 4 - 4\sin^2\theta + 7 \sin^2\theta$

$x - 4 = 3 \sin^2\theta$

So far so good...

$\int_4^7 \frac{1}{\sqrt{(x - 4)(7 - x)}}d\theta$

$\int_4^7 \frac{1}{\sqrt{(3\sin^2\theta)(3\cos^2\theta)}}d\t heta$

Erase this step: if you already changed variables you must already change dx by $d\theta$ and also the integral's limits

$\int_0^{90} \frac{6\cos\theta\sin\theta}{\sqrt{(3\sin^2\theta) (3\cos^2\theta)}}d\theta$

!!! Haven't you been told that you can not use degrees when dealing with differential/integral calculus and trigonometric

function but you must shift to radians?! The limits must be $\int\limits_0^{\pi/2}$

$\int_0^{90} \frac{3\sin 2\theta}{\sqrt{9\cos^2\theta - 9\cos^4\theta}}$

$\int_0^{90} \frac{3\sin2\theta}{3\sqrt{cos^2\theta - \cos^4\theta}}d\theta$

$\int_0^{90} \frac{\sin 2\theta}{\sqrt{\cos^2\theta(1 - cos^2\theta)}}d\theta$

$\int_0^{90} \frac{2\sin\theta}{cos\theta\sqrt{(1 - cos^2\theta)}}d\theta$

$2 \sin\theta \int_0^{90} \frac{1}{1 - \cos^2\theta}d\theta$

$\big[2\sin\theta \cdot \sin^{-1}(\cos\theta\big]_0^{90}$

$[2\cdot 1 \cdot \sin^{-1}(0)] - [0 \cdot 0]$

$0$

I'm afraid you messed up big time with the easiest part of the solution, after dealing nicely with the hardest part!

You already had $\int\limits_0^{\pi/2} \frac{6\cos\theta\sin\theta}{\sqrt{(3\sin^2\theta) (3\cos^2\theta)}}d\theta=\int\limits_0^{\pi/2}\frac{6\cos\theta\sin\theta}{3\cos\theta\sin\the ta}d\theta=2\int\limits_0^{\pi/2}d\theta=\pi$...as simple as that!

Tonio