1) Evaluate using a trigonometric substitution: a) int x^3sqrt9-x^2 dx b) int1...

Question:

1) Evaluate using a trigonometric substitution:

{eq}\displaystyle \quad a)\ \int x^3\sqrt{9-x^2}\ dx\\[4ex] \displaystyle \quad b)\ \int\frac{1}{x^2\sqrt{x^2+4}}\ dx\\[4ex] {/eq}

2) Evaluate using partial fractions.

{eq}\displaystyle \quad a)\ \int\frac{1}{x^2(x-1)^2}\ dx \\[4ex] \displaystyle \quad b)\ \int\frac{x^2-x+6}{x^3+3x}\ dx\\[4ex] {/eq}

3) Evaluate.

{eq}\displaystyle \quad a)\ \int\frac{x}{\sqrt{4-x^4}}\ dx \\[4ex] \displaystyle \quad b)\ \int\frac{x^3}{x^2+4}\ dx,\ Hint:divide\\[4ex] \displaystyle \quad c)\ \int\frac{\sqrt{x}}{x-4}\ dx,\ Hint:u=\sqrt{x} {/eq}

Integration

Integrals can be evaluate with a variety of methods, including trigonometric substitution, partial fractions, and basic integration formulas. We'll use all of these methods to solve the given problems.

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1) Evaluate using trigonometric substitution:

a) {eq}\displaystyle \int x^3 \sqrt{9-x^2} dx {/eq}

Let {eq}\displaystyle x = 3\sin \theta \\ dx =...

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How to Use Trigonometric Substitution to Solve Integrals

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Chapter 13 / Lesson 12
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Trigonometric substitutions can be useful by plugging in a function of a variable, thus simplifying the calculation of an integral. Learn how to solve integrals using substitution, tables, by parts, and Riemann Sums through a variety of examples.


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