7.2.1 integrate trig with radical

In summary, the conversation discusses solving an integral using substitution and resetting the limits of integration. The double angle identity is also used to simplify the integral. The final answer is $\frac{\sqrt{2}}{2}$.
  • #1
karush
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  • #2
\(\displaystyle \int_0^{\pi/4} \sqrt{1+\cos(4x)} \, dx\)

$u=2x \implies du = 2 \, dx$

\(\displaystyle \frac{1}{2} \int_0^{\pi/4} \sqrt{1+\cos[2(2x)]} \cdot 2 \, dx\)

substitute and reset the limits of integration ...

\(\displaystyle \frac{1}{2} \int_0^{\pi/2} \sqrt{1+\cos(2u)} \, du\)

note the double angle identity, $\cos(2u) = 2\cos^2{u}-1 \implies 1+\cos(2u) = 2\cos^2{u}$

\(\displaystyle \frac{1}{2} \int_0^{\pi/2} \sqrt{2\cos^2{u}} \, du\)

note $\sqrt{2\cos^2{u}} = \sqrt{2} \cdot \sqrt{\cos^2{u}} = \sqrt{2} \cdot |\cos{u}| = \sqrt{2} \cdot \cos{u}$ since $\cos{u} \ge 0$ in quadrant I.

\(\displaystyle \frac{\sqrt{2}}{2} \int_0^{\pi/2}\cos{u} \, du\)

$\dfrac{\sqrt{2}}{2} \bigg[\sin{u} \bigg]_0^{\pi/2} = \dfrac{\sqrt{2}}{2} (1-0) = \dfrac{\sqrt{2}}{2}$
 
  • #3
ok I see now it was resetting the limits because of substitution

Mahalo
 

Related to 7.2.1 integrate trig with radical

1. What is the purpose of integrating trig with radical?

The purpose of integrating trigonometric functions with radical expressions is to simplify and solve complex mathematical equations involving both trigonometric functions and radical expressions. This can help in solving real-world problems in fields such as engineering, physics, and mathematics.

2. What are the basic trigonometric functions used in integration with radical?

The basic trigonometric functions used in integration with radical are sine, cosine, and tangent. These functions are commonly used in trigonometry and are essential in solving equations involving both trigonometric and radical expressions.

3. How do you integrate trig functions with radical expressions?

To integrate trigonometric functions with radical expressions, you can use substitution or integration by parts. In substitution, you replace the radical expression with a new variable and use the appropriate trigonometric identity to simplify the equation. In integration by parts, you split the equation into two parts and integrate each part separately.

4. What are some common applications of integrating trig functions with radical?

Integrating trigonometric functions with radical expressions has various applications in fields such as physics, engineering, and mathematics. Some common applications include finding the area under a curve, calculating the volume of a solid, and solving differential equations.

5. Are there any tips for solving integration problems involving trig functions and radical expressions?

Some tips for solving integration problems involving trigonometric functions and radical expressions include identifying the appropriate substitution or integration by parts method, using trigonometric identities to simplify the equation, and checking your answer by differentiating the integrated equation. It is also helpful to practice various integration problems to improve your skills.

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