Question Statement

Evaluate the integral:

$\int \mathrm{cosec}11x\tan 11xdx$

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Background and Explanation

This integral requires converting trigonometric functions to their sine and cosine definitions to simplify the integrand. You should be familiar with basic trigonometric identities and the standard integral formula for $\sec x$.

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Solution

We begin by expressing the integrand in terms of sine and cosine functions to identify any simplifications.

First, recall that $\mathrm{cosec}\theta =\frac{1}{\sin \theta }$ and $\tan \theta =\frac{\sin \theta }{\cos \theta }$. Applying these to our integral:

$\begin{array}{rl}I& =\int \mathrm{cosec}11x\tan 11xdx\\ & =\int \frac{1}{\sin 11x}\cdot \frac{\sin 11x}{\cos 11x}dx\end{array}$

Notice that the $\sin 11x$ terms in the numerator and denominator cancel each other:

$\begin{array}{rl}I& =\int \frac{1}{\cos 11x}dx\\ & =\int \sec 11xdx\end{array}$

Now we use the substitution method to handle the argument $11x$. Let:

$11x=t$

Differentiating both sides with respect to $x$:

$11dx=dt$

Solving for $dx$:

$dx=\frac{1}{11}dt$

Substituting back into our integral:

$\begin{array}{rl}I& =\int \sec t\cdot \frac{1}{11}dt\\ & =\frac{1}{11}\int \sec tdt\end{array}$

We apply the standard integral formula $\int \sec xdx=\ln |\sec x+\tan x|+C$:

$\begin{array}{rl}I& =\frac{1}{11}\ln |\sec t+\tan t|+c\end{array}$

Finally, we back-substitute $t=11x$ to express the answer in terms of the original variable:

$I=\frac{1}{11}\ln |\sec 11x+\tan 11x|+c$

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Key Formulas or Methods Used

• Trigonometric identities: $\mathrm{cosec}x=\frac{1}{\sin x}$, $\tan x=\frac{\sin x}{\cos x}$, and $\sec x=\frac{1}{\cos x}$
• Algebraic simplification: Cancellation of common factors in numerator and denominator
• Integration by substitution: $u$-substitution (or $t$-substitution) to handle composite functions
• Standard integral: $\int \sec xdx=\ln |\sec x+\tan x|+C$

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Summary of Steps

1. Convert to sine/cosine form: Rewrite $\mathrm{cosec}11x$ as $\frac{1}{\sin 11x}$ and $\tan 11x$ as $\frac{\sin 11x}{\cos 11x}$
2. Simplify the integrand: Cancel $\sin 11x$ to obtain $\int \sec 11xdx$
3. Substitute: Let $t=11x$, which gives $dx=\frac{dt}{11}$
4. Integrate: Apply the standard formula $\int \sec tdt=\ln |\sec t+\tan t|+C$ with the constant factor $\frac{1}{11}$
5. Back-substitute: Replace $t$ with $11x$ to get the final answer $\frac{1}{11}\ln |\sec 11x+\tan 11x|+C$