Evaluate the following integral. sin^2 3x dx a. fraction 1 4 x - fraction 1 12 sin 6x + c b....


Evaluate the following integral.

{eq}\displaystyle \int \sin^2 (3x) \ dx {/eq}

{eq}\text{a.} \ \dfrac{1}{4} x - \dfrac{1}{12} \sin (6x) + c \\ \text{b.} \ \dfrac{1}{2} x - \dfrac{1}{6} \sin (3x) + c \\ \text{c.} \ \dfrac{1}{2} x - \dfrac{1}{12} \sin (6x) + c \\ \text{d.} \ \dfrac{1}{2} x - \dfrac{1}{6} \sin (6x) + c {/eq}

Indefinite Integral:

The substitution method or the standard formula can be used for the indefinite integral of the trigonometric function. For example, the following trigonometric functions are useful.

{eq}\sin^{2}\theta = \dfrac{1-\cos2\theta}{2} {/eq}

Answer and Explanation: 1

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Let the required integral be {eq}I {/eq} then we have:

{eq}\begin{align*} I & = \displaystyle \int \sin^{2} (3x) \ dx \\[0.3cm] & = \displaystyle...

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Indefinite Integrals as Anti Derivatives


Chapter 12 / Lesson 11

Indefinite integrals, an integral of the integrand which does not have upper or lower limits, can be used to identify individual points at specific times. Learn more about the fundamental theorem, use of antiderivatives, and indefinite integrals through examples in this lesson.

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