# Definite Integration with Trigonometric Substitution

I'm working on a question that involves using trigonometric substitution on a definite integral that will later use u substitution but I am not sure how to go ahead with this.

$$\int_1^2\frac1{x^2\sqrt{4x^2+9}}dx$$

My first step was to use $\sqrt{a^2+x^2}$ as $x=a\tan\theta$ to get...

$$2x=3\tan\theta :x=\frac32\tan\theta$$ $$dx=\frac32\sec^2\theta$$

Substituting:

$$\int\frac{\frac32\sec^2\theta}{\frac94\tan^2\theta\sqrt{9\tan^2\theta+9}}$$

The problem here is how do I change the limit it goes to?

$$\frac43=\tan\theta$$ and $$\frac23=\tan\theta$$

$$\frac29\int_{\tan^-1(\frac23)}^{\tan^-1(\frac43)}\frac{\sec^2\theta}{\tan^2\theta\sqrt{9\sec^2\theta}}d\theta$$

$$=\frac29\int_{\tan^-1(\frac23)}^{\tan^-1(\frac43)}\frac{\sec\theta}{\tan^2\theta}d\theta$$

$$=\frac29\int_{\tan^-1(\frac23)}^{\tan^-1(\frac43)}\frac{\cos\theta}{\sin^2\theta}d\theta$$

Now $u=\sin\theta$ so $du=\cos\theta d\theta$

$$=\frac29\int_{?}^{?}\frac{1}{u^2}du$$

This is where I am stuck now...

• Do you need to know those values? Maybe you can just live with them as values of inverse trig functions. After all, your answer is likely to involve other trig functions evaluated at $\theta$.
– lulu
Commented Jul 21, 2015 at 22:19
• No problem! Denote by $\theta_1$ and $\theta_2$ the angle such that $\tan\theta_1=\frac{4}{3}$ and $\tan\theta_2=\frac{2}{3}$. In the end you will find expressions involving $\tan$ and $\sec$. Commented Jul 21, 2015 at 22:20
• Sorry could you explain this a bit more? Do you mean for me to continue my integration from $\arctan\frac23$ to $\arctan\frac43$ Commented Jul 21, 2015 at 22:22
• Compute the indefinite integral and express the result in function of $\tan\theta$. Commented Jul 21, 2015 at 22:24
• @Panthy Yes. Exactly. If you know the value of one trig function at an angle it is generally pretty easy to find the value of the others.
– lulu
Commented Jul 21, 2015 at 22:30

There was a typo in the current post. After enforcing the substitution $2x=3\tan \theta$, the integral ought to read

\begin{align}I&=\int_{\arctan(2/3)}^{\arctan(4/3)}\frac{\frac32 \sec^2\theta}{\frac94 \tan^2\theta\sqrt{9\tan^2\theta+9}}d\theta\\\\ &=\frac29 \int_{\arctan(2/3)}^{\arctan(4/3)}\frac{ \sec^2\theta}{ \tan^2\theta\,\sec \theta}d\theta\\\\ &=\frac29 \int_{\arctan(2/3)}^{\arctan(4/3)}\frac{ \sec^2\theta}{ \tan^2\theta\,\sec \theta}d\theta\\\\ &=\frac29 \int_{\arctan(2/3)}^{\arctan(4/3)}\cot \theta \csc \theta d\theta\\\\ &=\frac29 \left.(-\csc \theta)\right|_{\arctan(2/3)}^{\arctan(4/3)}\\\\ &=\frac{\sqrt{13}}{9}-\frac{5}{18}\end{align}

NOTES:

Remark 1: When making a substitution of variables in a definite integral, the limits of integration change accordingly. In this example, the substitution was $x=\frac32 \tan \theta$. When $x=1$ at the lower limit, $\tan \theta =\frac23\implies \theta =\arctan(2/3)$. Similarly, when $x=2$ at the upper limit, $\tan \theta =\frac43\implies \theta =\arctan(4/3)$.

Remark 2:
To evaluate $\sin (\arctan(2/3))$, we recall that the arctangent is an angle whose tangent is $2/3$. A picture sometimes facilitates the analysis wherein we draw a right triangle with vertical side of length $2$ and horizontal side of length $3$ forming a right angle.

Note that the angle the hypotenuse makes with the horizontal side is $\arctan(2/3)$. Inasmcuh as the hypotenuse is of length $\sqrt{2^2+3^2}=\sqrt{13}$, we see $\sin(\arctan(2/3))=\frac{2}{\sqrt{13}}$ and thus $\csc (\arctan(2/3))=\frac{\sqrt{13}}{2}$.

• will continue from this point and update you! thanks Commented Jul 21, 2015 at 22:32
• i updated my answer but still confused Commented Jul 21, 2015 at 22:43
• OK ... posted the full solution to assist. Please let me know how I can improve the answer. I just want to give you the best answer I can. Commented Jul 21, 2015 at 22:50
• is the way I did it wrong? im not so confident dealing with csc sec and stuff but either way im reallly not sure how you evaluate $(-csc\theta)_{arctan(\frac23)}^{arctan(\frac43)}$ Commented Jul 21, 2015 at 22:52
• The arctangent gives an angle whose tangent is the argument. So, if an angle has a tangent of $2/3$ then its sine is $\sqrt{2^2+3^2}=\sqrt{13}$. Likewise, the sine of an angle whose tangent is $4/3$ is $\sqrt{4^2+3^2}=5$. How else can I improve the answer? Commented Jul 21, 2015 at 23:00

\begin{align*}\int\frac{\frac32\sec^2\theta\,\mathrm d\mkern1.5mu\theta}{\frac94\tan^2\theta\sqrt{9\tan^2\theta+9}}&=\frac29\int\frac{\mathrm d\mkern1.5mu\theta}{\sin^2\theta\sqrt{1+\tan^2\theta}}=\frac29\int\frac{\lvert\cos\theta\rvert\,\mathrm d\mkern1.5mu\theta}{\sin^2\theta}\\[1ex] &=\frac29\int\frac{\cos\theta\,\mathrm d\mkern1.5mu\theta}{\sin^2\theta}\qquad\text{since}\enspace 0\le\theta<\dfrac\pi2\\[1ex] &=-\frac2{9\sin\theta} \end{align*} Some trigonometry will let you determine the bounds for $\sin\theta\;$ from the bounds for $\tan\theta$: since $0\le\theta<\dfrac\pi2$, we have: $\cos\theta=\dfrac1{\sqrt{1+\tan^2\theta}}$, hence $$\sin\theta=\tan\theta\cos\theta=\frac{\tan\theta}{\sqrt{1+\tan^2\theta}}=\frac{\cfrac{2x}3}{\sqrt{1+\cfrac{4x^2}9}}=\frac{2x}{\sqrt{4x^2+9}}$$ so that the indefinite integral is: $$-\frac{\sqrt{4x^2+9}}{9x}.$$

• There appears to be an error of omission. The secant squares term disappeared in the second integral. Commented Jul 21, 2015 at 22:52
• It's been replaced with $\cos^2\theta$ in the denominator; Commented Jul 21, 2015 at 22:54
• Why the downvote? Commented Jul 21, 2015 at 22:55
• OK. Yes. But the integrand is missing the term in the denominator, namely $x^2=\frac94 \tan^2 \theta$. Commented Jul 21, 2015 at 22:56
• By the way, I was not the down voter. In fact, I very very rarely down vote. Rather, I like to neutralize down votes by giving an up vote. Commented Jul 21, 2015 at 22:57

$$\int_1^2\frac1{x^2\sqrt{4x^2+9}}dx = \int_a^b\frac{\frac{2}{3}\sec^2\theta}{\frac{9}{4}\tan^2\theta\sqrt{9\tan^2\theta+9}}d\theta$$

where $a = \tan^{-1}\frac{2}{3}$ and $b = \tan^{-1}\frac{4}{3}$

$$\int_a^b\frac{2\sec\theta}{9\tan^2\theta}d\theta = \left(-\frac{2\text{cosec}\theta}{9} \right)_a^b$$

Hint: $\text{cosec}\left(\tan^{-1}\frac{4}{3}\right) = \frac{5}{4}$, $\text{cosec}\left(\tan^{-1}\frac{2}{3}\right) = \frac{\sqrt{13}}{2}$

EDIT: let $\tan^{-1}\frac{4}{3}=d$

$$\frac{4}{3} = \tan d$$ $$\frac{4}{3} = \frac{\sin d}{\cos d}$$ $$\frac{1}{\sin d} = \text{cosec}\ d = \frac{3}{4}\sec d$$

$$\sec^2d = \tan^2 d + 1 = 1+ \left(\frac{4}{3}\right)^2 = \frac{25}{9}$$ $$\sec d =\frac{5}{3}$$

$$\text{cosec}\ d = \frac{3}{4}\times \frac{5}{3} =\frac{5}{4}$$

You can do the same for the second one.

• how do you know that? Commented Jul 21, 2015 at 22:47
• could you please elaborate im new to that kind of thing... sorry if i sound stupid Commented Jul 21, 2015 at 22:54
• I'll explain further in my solution. Commented Jul 21, 2015 at 22:54
• @Panthy, I hope the explanation helps. Commented Jul 21, 2015 at 23:10