Math Problem Statement

Universal GPT is designed to tackle a comprehensive array of questions across various subjects, ranging from mathematics and science to literature, history, and beyond. This AI excels at interpreting and solving complex problems, providing not only solutions but also detailed, step-by-step explanations to ensure a thorough understanding of the processes involved. It covers diverse subjects, including algebra, geometry, trigonometry, calculus, physics, chemistry, biology, literature analysis, historical events, and more, demonstrating exceptional reasoning capabilities and attention to detail. For photo-based inquiries, Universal GPT efficiently extracts and analyzes content, delivering precise and accurate answers. When addressing text-based questions, it applies rigorous analytical methods to offer insightful explanations and accurate results. The tool is designed to enhance users' understanding through clear, educational explanations, maintaining a formal and supportive tone to foster learning. Universal GPT seeks clarifications when needed to ensure precision, making it a reliable and comprehensive problem-solving companion for students, educators, professionals, and anyone with a thirst for knowledge: ∫0π/4 sin 2x dx

Solution

To solve the integral 0π/4sin(2x)dx\int_{0}^{\pi/4} \sin(2x) \, dx, we will use the following steps:

  1. Substitution: Let u=2xu = 2x. Then du=2dxdu = 2 \, dx, which implies dx=12dudx = \frac{1}{2} \, du.

  2. Changing the Limits: When x=0x = 0: u=20=0u = 2 \cdot 0 = 0

    When x=π4x = \frac{\pi}{4}: u=2π4=π2u = 2 \cdot \frac{\pi}{4} = \frac{\pi}{2}

  3. Rewrite the Integral: Substitute uu and dxdx into the integral: 0π/4sin(2x)dx=0π/2sin(u)12du=120π/2sin(u)du\int_{0}^{\pi/4} \sin(2x) \, dx = \int_{0}^{\pi/2} \sin(u) \cdot \frac{1}{2} \, du = \frac{1}{2} \int_{0}^{\pi/2} \sin(u) \, du

  4. Evaluate the Integral: The integral of sin(u)\sin(u) is cos(u)-\cos(u): 120π/2sin(u)du=12[cos(u)]0π/2\frac{1}{2} \int_{0}^{\pi/2} \sin(u) \, du = \frac{1}{2} \left[ -\cos(u) \right]_{0}^{\pi/2}

  5. Apply the Limits: Evaluate cos(u)-\cos(u) from 00 to π/2\pi/2: [ \frac{1}{2} \left[ -\cos\left( \frac{\pi}{2} \right) + \cos(0) \right] = \frac{1}{2} \left[ -0 + 1 \right] = \frac{1}{2} \left[ 1 \right] = \frac{1}{2} ]

So, the value of the integral 0π/4sin(2x)dx\int_{0}^{\pi/4} \sin(2x) \, dx is 12\boxed{\frac{1}{2}}.

Would you like more details or have any questions about this solution? Here are some related questions for further practice:

  1. Evaluate 0π/6cos(3x)dx\int_{0}^{\pi/6} \cos(3x) \, dx.
  2. Find the antiderivative of sin(5x)\sin(5x).
  3. Determine 0π/2sin(x)dx\int_{0}^{\pi/2} \sin(x) \, dx.
  4. Calculate 0πcos(2x)dx\int_{0}^{\pi} \cos(2x) \, dx.
  5. Solve 0π/4sin(x)dx\int_{0}^{\pi/4} \sin(x) \, dx.
  6. Evaluate 0π/3cos(4x)dx\int_{0}^{\pi/3} \cos(4x) \, dx.
  7. Find 02πsin(x)dx\int_{0}^{2\pi} \sin(x) \, dx.
  8. Determine 0π/6sin(2x)dx\int_{0}^{\pi/6} \sin(2x) \, dx.

Tip: When dealing with trigonometric integrals, using substitution can simplify the process significantly. Always remember to change the limits of integration when you make a substitution.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Calculus
Trigonometry

Formulas

Substitution method
Integral of sine function

Theorems

-

Suitable Grade Level

Advanced High School and Above