Math Problem Statement

Integrate number 1-9 using step by step process do not use integration by parts. Box the final answer

Integrate number 1-9 using step by step process do not use integration by parts. Box the final answer

Solution

Let's walk through each integration step by step, using the most common methods available, without using integration by parts. We'll box the final answers as you requested.

1. sin4(x)dx\int \sin^4(x) \, dx

We start by expressing sin4(x)\sin^4(x) as a combination of lower powers of sine using the power reduction formula:

sin4(x)=(sin2(x))2=1cos(2x)21cos(4x)2\sin^4(x) = \left( \sin^2(x) \right)^2 = \frac{1 - \cos(2x)}{2} \cdot \frac{1 - \cos(4x)}{2}

Now, we simplify the expression:

sin4(x)=14(1cos(2x))(1cos(4x))\sin^4(x) = \frac{1}{4} (1 - \cos(2x)) (1 - \cos(4x))

Next, expand the product:

=14(1cos(2x)cos(4x)+cos(6x))= \frac{1}{4} (1 - \cos(2x) - \cos(4x) + \cos(6x))

Now integrate term by term:

sin4(x)dx=14(1dxcos(2x)dxcos(4x)dx+cos(6x)dx)\int \sin^4(x) \, dx = \frac{1}{4} \left( \int 1 \, dx - \int \cos(2x) \, dx - \int \cos(4x) \, dx + \int \cos(6x) \, dx \right)

=14(xsin(2x)2sin(4x)4+sin(6x)6)+C= \frac{1}{4} \left( x - \frac{\sin(2x)}{2} - \frac{\sin(4x)}{4} + \frac{\sin(6x)}{6} \right) + C

Thus, the final answer is:

14cos(4x)+112cos(4x)+C\boxed{ \frac{1}{4} \cos(4x) + \frac{1}{12} \cos(4x) + C }

2. cos4(x)dx\int \cos^4(x) \, dx

Using a similar approach for cos4(x)\cos^4(x):

cos4(x)=(cos2(x))2=1+cos(2x)21+cos(4x)2\cos^4(x) = \left( \cos^2(x) \right)^2 = \frac{1 + \cos(2x)}{2} \cdot \frac{1 + \cos(4x)}{2}

Now expand the product:

cos4(x)=14(1+cos(2x))(1+cos(4x))\cos^4(x) = \frac{1}{4} (1 + \cos(2x)) (1 + \cos(4x))

After expanding:

=14(1+cos(2x)+cos(4x)+cos(6x))= \frac{1}{4} (1 + \cos(2x) + \cos(4x) + \cos(6x))

Now integrate term by term:

cos4(x)dx=14(x+sin(2x)2+sin(4x)4+sin(6x)6)+C\int \cos^4(x) \, dx = \frac{1}{4} \left( x + \frac{\sin(2x)}{2} + \frac{\sin(4x)}{4} + \frac{\sin(6x)}{6} \right) + C

Thus, the final answer is:

12x14sin(2x)+14sin(2x)+C\boxed{ \frac{1}{2} x - \frac{1}{4} \sin(2x) + \frac{1}{4} \sin(2x) + C }

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Math Problem Analysis

Mathematical Concepts

Integral Calculus
Trigonometric Integrals

Formulas

Power reduction formula for trigonometric functions
Standard integrals for sin^n(x) and cos^n(x)
Integration of sec^2(x), tan(x), etc.

Theorems

Power Reduction Formula
Basic Integration Techniques

Suitable Grade Level

Grades 11-12