T1.14 Integral: trigonometric u-substitution

In summary, Trigonometric u-substitution is a technique used to simplify integrals involving trigonometric functions by substituting a trigonometric expression with a new variable, u. It can be used when the integral involves a product or composition of trigonometric functions, or when a trigonometric function is raised to a power. Two common formulas for u substitution are u = sinx and u = cosx. It is only applicable for integrals involving trigonometric functions and it is important to choose the appropriate u-value and use trigonometric identities to simplify the integrand for effective use.
  • #1
karush
Gold Member
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$\tiny{2214.t1.14}$
$\text{Evaluate the Integral:}$
\begin{align*}\displaystyle
I_{14}&=\int \frac{12\tan^2x \sec^2 x}{(4+\tan^3x)^2} \, dx \\
\textit{Use U substitution}&\\
u&=4+\tan^3x\\
\, \therefore dx& =\dfrac{1}{3\sec^2\left(x\right)\tan^2\left(x\right)}\,du\\
&=4 \int\frac{1}{u^2}\,du\\
&=4\left[-\dfrac{1}{u} \right]\\
\textit{Back substitute $u=4+\tan^3x$}\\
I_{14}&=-\frac{4}{4+\tan^3x}+C
\end{align*}

ok just seeing if this is correct
and suggestions
 
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  • #2
Re: t1.14 Integral: trigonometic u-substitution

My suggestion, again, is not to use so many abbreviations in thread titles. (Yes)
 

FAQ: T1.14 Integral: trigonometric u-substitution

1. What is the purpose of using trigonometric u-substitution in T1.14 Integral?

Trigonometric u-substitution is a technique used to simplify integrals involving trigonometric functions. It involves substituting a trigonometric expression with a new variable, u, to make the integral easier to solve.

2. How do I know when to use trigonometric u-substitution?

You can use trigonometric u-substitution when the integral involves a product of trigonometric functions, a composition of trigonometric functions, or when a trigonometric function is raised to a power.

3. Do I need to memorize any formulas for trigonometric u-substitution?

Yes, there are two common formulas used in trigonometric u-substitution: u = sinx and u = cosx. These formulas can be used depending on the trigonometric function present in the integral.

4. Can I use trigonometric u-substitution for any type of integral?

No, trigonometric u-substitution is only applicable for integrals involving trigonometric functions. It cannot be used for other types of integrals such as polynomial or exponential functions.

5. Are there any tips for using trigonometric u-substitution effectively?

Yes, it is important to choose the appropriate u-value to substitute in order to simplify the integral. It is also helpful to use trigonometric identities to simplify the integrand before applying u-substitution.

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