Trigonometric integrals Evaluate the following integrals. /2sin 2x cos 2r dx/4

We have to evaluate the integral ∫π4π2sin22xcos32xdx

Answered: /2 sin 2x cos 2r dx /4

Math

Calculus

/2 sin 2x cos 2r dx /4

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter6: The Trigonometric Functions

Section6.6: Additional Trigonometric Graphs

Problem 67E

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Concept explainers

Riemann Sum

Riemann Sums is a special type of approximation of the area under a curve by dividing it into multiple simple shapes like rectangles or trapezoids and is used in integrals when finite sums are involved. Figuring out the area of a curve is complex hence this method makes it simple. Usually, we take the help of different integration methods for this purpose. This is one of the major parts of integral calculus.

Riemann Integral

Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.

Question

Trigonometric integrals Evaluate the following integrals.

With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.

Expert Solution

Step 1

We have to evaluate the integral

$\int_{\frac{π}{4}}^{\frac{π}{2}} \sin^{2} 2 x \cos^{3} 2 x d x$

Step 2

$\int_{\frac{π}{4}}^{\frac{π}{2}} \sin^{2} 2 x \cos^{3} 2 x d x = \frac{1}{4} \int_{\frac{π}{4}}^{\frac{π}{2}} {(2 \sin 2 x \cos 2 x)}^{2} \cos 2 x d x = \frac{1}{4} \int_{\frac{π}{4}}^{\frac{π}{2}} \sin^{2} 4 x \cos 2 x d x W e c a n w r i t e \sin^{2} 4 x = 1 - \cos^{2} 4 x = \frac{1}{4} \int_{\frac{π}{4}}^{\frac{π}{2}} (1 - \cos^{2} 4 x) \cos 2 x d x = \frac{1}{4} \int_{\frac{π}{4}}^{\frac{π}{2}} \cos 2 x d x - \frac{1}{4} \int_{\frac{π}{4}}^{\frac{π}{2}} \cos^{2} 4 x \cos 2 x d x = \frac{1}{4} \sin 2 x {|^{\frac{π}{2}}}_{\frac{π}{4}} - \frac{1}{8} \int_{\frac{π}{4}}^{\frac{π}{2}} \cos 8 x \cos 2 x + \cos 2 x$

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