Math Problem Statement

𝜋/4 sin(t) cos2(t) dt

Solution

You appear to be asking for an integral involving the given terms. Let me interpret the integral as follows:

π/4sin(t)cos2(t)dt\int_{\pi/4} \sin(t) \cos^2(t) \, dt

To solve this:

Step 1: Simplify the integral

Using the trigonometric identity: cos2(t)=1sin2(t)\cos^2(t) = 1 - \sin^2(t) The integral becomes: π/4sin(t)(1sin2(t))dt\int_{\pi/4} \sin(t) (1 - \sin^2(t)) \, dt

Step 2: Use substitution

Let: u=sin(t)so thatdu=cos(t)dtu = \sin(t) \quad \text{so that} \quad du = \cos(t) \, dt

Change the limits of integration for uu:

  • When t=π/4t = \pi/4, u=sin(π/4)=22u = \sin(\pi/4) = \frac{\sqrt{2}}{2}.

Thus, the integral becomes: sin(t)(1sin2(t))dt=2/2uu(1u2)du\int \sin(t) (1 - \sin^2(t)) \, dt = \int_{\sqrt{2}/2}^{u} u(1 - u^2) \, du

Step 3: Expand and integrate

Expand u(1u2)u(1 - u^2): 2/2u(1u2)du=2/2(uu3)du\int_{\sqrt{2}/2} u(1 - u^2) \, du = \int_{\sqrt{2}/2} (u - u^3) \, du

Now integrate each term: udu=u22,u3du=u44\int u \, du = \frac{u^2}{2}, \quad \int u^3 \, du = \frac{u^4}{4}

Thus: 2/2u(1u2)du=[u22u44]2/21\int_{\sqrt{2}/2} u(1 - u^2) \, du = \left[\frac{u^2}{2} - \frac{u^4}{4} \right]_{\sqrt{2}/2}^1

Step 4: Evaluate the limits

First, calculate at u=1u = 1: 122144=1214=14\frac{1^2}{2} - \frac{1^4}{4} = \frac{1}{2} - \frac{1}{4} = \frac{1}{4}

Now, calculate at u=22u = \frac{\sqrt{2}}{2}: (22)22(22)44=2424164=14116=416116=316\frac{\left(\frac{\sqrt{2}}{2}\right)^2}{2} - \frac{\left(\frac{\sqrt{2}}{2}\right)^4}{4} = \frac{\frac{2}{4}}{2} - \frac{\frac{4}{16}}{4} = \frac{1}{4} - \frac{1}{16} = \frac{4}{16} - \frac{1}{16} = \frac{3}{16}

Subtract the results: 14316=416316=116\frac{1}{4} - \frac{3}{16} = \frac{4}{16} - \frac{3}{16} = \frac{1}{16}

Final Answer:

π/4sin(t)cos2(t)dt=116\int_{\pi/4} \sin(t) \cos^2(t) \, dt = \frac{1}{16}


Do you need further details or clarifications?

Expanding Questions:

  1. Would you like to verify each substitution step?
  2. Do you want a graph illustrating this integral geometrically?
  3. Shall I explain the trigonometric identity cos2(t)=1sin2(t)\cos^2(t) = 1 - \sin^2(t) in detail?
  4. Would you like to explore alternative methods to solve this integral?
  5. Are you interested in practical applications of such integrals?

Tip:

Always double-check the limits of integration after substitution to avoid errors in definite integrals!

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Math Problem Analysis

Mathematical Concepts

Integration
Trigonometry
Substitution
Definite Integrals

Formulas

cos²(t) = 1 - sin²(t)
∫ u(1 - u²) du

Theorems

Integration by Substitution
Fundamental Theorem of Calculus

Suitable Grade Level

Grades 11-12