Solve U-Sub Trig Integrals: Tan^5(3x) Sec^2(3x)

• Knissp
In summary, the conversation discusses the process of finding the integral of tan^5(3x) sec^2(3x) dx, and the difference between the answer obtained and the one given by a calculator. It is suggested to verify the answer by differentiating it, and the process of doing so is shown. The final result is that the obtained answer is equivalent to the one given by the calculator.
Knissp

Homework Statement

$$\int tan^5(3x) sec^2(3x) dx$$

The Attempt at a Solution

$$u = tan(3x)$$
$$du = 3 sec^2(3x) dx$$
$$du/3 = sec^2(3x) dx$$
$$\int tan^5(3x) sec^2(3x) dx$$
$$= \int 1/3 u^5 du$$
$$= u^6/18 + C$$
$$= tan^6(3x)/18 + C$$

The calculator says the integral should be:
$$\frac {1 - 3 sin^2(3x) cos^2(3x)}{18cos^6(3x)} + C$$

The answer I got does not differ from the calculator by a constant. Any help would be appreciated.

Knissp said:
$$= tan^6(3x)/18 + C$$

That's what I get too.

The calculator says the integral should be:
$$\frac {1 - 3 sin^2(3x) cos^2(3x)}{18cos^6(3x)} + C$$

The answer I got does not differ from the calculator by a constant.

Yes it does. Sometimes it's tricky to show the equivalence (up to a constant) of two antiderivatives of trigonometric integrands. It frequently can depend on using just the right combination of identities to show that they are in fact the same. My advice is to put down the calculator and check your integration the old fashioned way: take the derivative of your result and verify that you recover the integrand.

My answer can be easily verified by differentiating.

Differentiating the calculator's answer gives:

$$d/dx \frac {1 - 3 sin^2(3x) cos^2(3x)}{18cos^6(3x)} + C$$
$$d/dx (\frac{1}{18cos^6(3x)} - d/dx \frac{3 sin^2(3x) cos^2 (3x)}{18 cos^6(3x)}$$
$$tan(3x) sec^6(3x) - d/dx \frac{sin^2(3x)}{6cos^4(3x)}$$
$$\frac {sin(3x)}{cos^7(3x)} - d/dx \frac{tan^2(3x) sec^2(3x)}{6}$$
$$\frac {sin(3x)}{cos^7(3x)} - \frac {tan^2(3x) d/dx (sec^2(3x)}{6} - \frac {sec^2(3x) d/dx tan^2(3x)}{6}$$
$$\frac {sin(3x)}{cos^7(3x)} - \frac {tan^2(3x) 2 sec(3x) (sec(3x)tan(3x)) 3}{6} - \frac {sec^2(3x) 2 tan(3x) sec^2(3x) 3}{6}$$
$$\frac {sin(3x)}{cos^7(3x)} - tan^2(3x) sec(3x) (sec(3x)tan(3x)) - sec^2(3x) tan(3x) sec^2(3x)$$
$$\frac {sin(3x)}{cos^7(3x)} - tan^3(3x) sec^2(3x) - sec^4(3x) tan(3x)$$
$$\frac {sin(3x)}{cos^7(3x)} - \frac{sin^3(3x)}{cos^5(3x)} - \frac{sin(3x)}{cos^5(3x)}$$
$$\frac {sin(3x)}{cos^7(3x)} - \frac{sin^3(3x) cos^2(3x)}{cos^7(3x)} - \frac{sin(3x)cos^2(3x)}{cos^7(3x)}$$

The integrand was $$tan^5(3x) sec^2(3x) = \frac{sin^5(3x)}{cos^7(3x)}$$, so all I need to do now is show
$$sin^5(3x) = sin(3x) - sin^3(3x) cos^2(3x) - sin(3x)cos^2(3x)$$

$$sin(3x) - sin^3(3x) cos^2(3x) - sin(3x)cos^2(3x)$$
$$sin(3x) - sin(3x)cos^2(3x) - sin^3(3x) cos^2(3x)$$
$$sin(3x) (1 - cos^2(3x)) - sin^3(3x) cos^2(3x)$$
$$sin(3x) (sin^2(3x)) - sin^3(3x) cos^2(3x)$$
$$sin^3(3x) - sin^3(3x) cos^2(3x)$$
$$sin^3(3x) (1 - cos^2(3x))$$
$$sin^5(3x)$$

cool! Thanks!

1. What is a U-substitution?

A U-substitution is a technique used to simplify integrals by replacing a complicated expression with a new variable, u. This variable is chosen in such a way that it makes the integral easier to solve.

2. How do I know when to use U-substitution for trigonometric integrals?

To use U-substitution for trigonometric integrals, you should look for expressions involving trigonometric functions and their derivatives, such as tan(x), sec(x), cot(x), csc(x), etc. If the integrand contains a product of these functions, U-substitution can be a useful technique.

3. What are the steps to solve U-substitution trigonometric integrals?

The steps to solve U-substitution trigonometric integrals are:

1. Identify the expression or function that is complicated and choose a new variable, u, to replace it.

2. Calculate the derivative of u, du, and substitute it into the integral.

3. Rewrite the integral in terms of u.

4. Solve the integral in terms of u.

5. Substitute the original expression back in for u.

4. What is the solution to the integral tan^5(3x)sec^2(3x)dx?

The solution to this integral is ∫tan^5(3x)sec^2(3x)dx = (1/3)tan^6(3x) + C, where C is the constant of integration.

5. Can I use U-substitution for all trigonometric integrals?

No, U-substitution can only be used for certain types of trigonometric integrals. It is most useful for integrands involving products of trigonometric functions and their derivatives. Other techniques, such as trigonometric identities, may be more suitable for other types of trigonometric integrals.

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