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Calculus BTrigonometric Substitution Problems 1
Subscribe to Calculus B
Application of Differentiation 83:08
Indeterminate forms and LHospital Rule83:08
Application of Integration 93:13
Area between Curves46:49
Volumes46:24
Techniques of Integration
Trigonometric Integrals
Integration of Rational Functions by Partial Fractions
Strategy for Integration
Further Application of Integration 61:59
Arc length61:59
Area of a Surface of Revolution
Application to physics and Engineering
Parametric Equation and Polar Coordinates
Curves Defined by Parametric Equations
Calculus with Parametric curves
Areas and Lengths in Polar Coordinates52:41
Infinite Sequences and Series 391:39
Sequences47:45
Series58:56
The Integral Test and Estimates of Sums35:15
The Comparison Test30:46
Alternating Series30:52
Absolute Convergence and The Ratio and Root Tests62:59
Representations of Functions as Power Series44:25
Taylor and Maclaurin Series80:41
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Practice (Free)
اسأل مدرس
Evaluate \(\int \frac{x^{2}}{\sqrt{9-x^{2}}} d x\)
Case 1: \(\sqrt{a^{2}-x^{2}}\)
\(x=3 \sin \theta \quad-\frac{\pi}{2} \le \theta \le \frac{\pi}{2}\)
\(d x=3 \cos \theta d \theta\)
\(\sqrt{9-x^{2}}=3 \cos \theta\)
\(I= \int \frac{x^{2}}{\sqrt{2-x^{2}}} d x=\int \frac{(3 \sin \theta)^{2}}{3 \cos \theta} 3 \cos \theta d \theta\)
\(I=\int 9 \sin ^{2} \theta d \theta \quad \text { but } \sin ^{2} \theta=\frac{1}{2}[1-\cos (2 \theta)]\)
\(I=\int 9\left(\frac{1}{2}(1-\cos (2 \theta))\right) d \theta\)
\(=\frac{9}{2} \int(1-\cos 2 \theta) d \theta=\frac{9}{2}\left[\theta-\frac{1}{2} \sin (2\theta)\right]+c\)
\(* \sin (2 \theta)=2 \sin \theta \cos \theta\)
\(I=\frac{9}{2}\left[\theta-\frac{1}{2}[2 \sin \theta \cos \theta]\right] + c\)
\(=\frac{9}{2}[\theta-\sin \theta \cos \theta]+c\)
\(\sin \theta=\frac{x}{3}\)
\(I=\int \frac{x^{2}}{\sqrt{9-x^{2}}} d x\)
\(I=\int \frac{x^{2}}{\sqrt{9-x^{2}}} d x=\frac{9}{2}[\theta-\sin \cos \theta]+c\)
\(\theta=\sin ^{-1}\left(\frac{x}{3}\right), \cos \theta=\frac{\sqrt{9-x^{2}}}{3}\)
\(=\frac{9}{2}\left[\sin ^{-1}\left((\frac{x}{3})-\frac{x}{3} \frac{\sqrt{9-{x}^{2}}}{3}\right)\right]\)
\(=\frac{9}{2}\left[\sin ^{-1}\left(\frac{x}{3}\right)-\frac{x \sqrt{9-{x}^{2}}}{9}\right]\)
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