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Section 3
Trigonometric Substitution
Evaluate the integral using the indicated trigonometric substitution. Sketch and label the associated right triangle.
$ \displaystyle \int \frac{dx}{x^2 \sqrt{4 - x^2}} $ $ x = 2 \sin \theta $
$ \int \frac{ x^3 }{ \sqrt{x^2 + 4}\ } dx $$ x = 2 \tan \theta $
$ \displaystyle \int \frac{\sqrt{x^2 - 4}}{x}\ dx $ $ x = 2 \sec \theta $
Evaluate the integral.
$ \displaystyle \int \frac{x^2}{\sqrt{9 - x^2}}\ dx $
$ \displaystyle \int \frac{\sqrt{x^2 - 1}}{x^4}\ dx $
$ \displaystyle \int_0^3 \frac{x}{\sqrt{36 - x^2}}\ dx $
Evaluate the integral.$\int_{0}^{a} \frac{d x}{\left(a^{2}+x^{2}\right)^{3 / 2}}, \quad a>0$
$ \displaystyle \int \frac{dt}{t^2 \sqrt{t^2 - 16}} $
$ \displaystyle \int_2^3 \frac{dx}{(x^2 - 1)^{\frac{3}{2}}} $
$ \displaystyle \int_0^{\frac{2}{3}} \sqrt{4 - 9x^2}\ dx $
$ \displaystyle \int_0^{\frac{1}{2}} x \sqrt{1 - 4x^2}\ dx $
$ \displaystyle \int_0^2 \frac{dt}{\sqrt{4 + t^2}} $
$ \displaystyle \int \frac{\sqrt{x^2 - 9}}{x^3}\ dx $
$ \displaystyle \int_0^1 \frac{dx}{(x^2 + 1)^2} $
$ \displaystyle \int_0^a x^2 \sqrt{a^2 - x^2}\ dx $
$ \displaystyle \int_{\frac{\sqrt{2}}{3}}^{\frac{2}{3}} \frac{dx}{x^5 \sqrt{9x^2 - 1}} $
$ \displaystyle \int \frac{x}{\sqrt{x^2 - 7}}\ dx $
$ \displaystyle \int \frac{dx}{[(ax)^2 - b^2]^{\frac{3}{2}}} $
$ \displaystyle \int \frac{\sqrt{1 + x^2}}{x}\ dx $
$ \displaystyle \int \frac{x}{\sqrt{1 + x^2}}\ dx $
$ \displaystyle \int_0^{0.6} \frac{x^2}{\sqrt{9 - 25x^2}}\ dx $
$ \displaystyle \int_0^1 \sqrt{x^2 + 1}\ dx $
$ \displaystyle \int \frac{dx}{\sqrt{x^2 + 2x + 5}} $
$ \displaystyle \int_0^1 \sqrt{x - x^2}\ dx $
$ \displaystyle \int x^2 \sqrt{3 + 2x - x^2}\ dx $
$ \displaystyle \int \frac{x^2}{(3 + 4x - 4x^2)^{\frac{3}{2}}}\ dx $
$ \displaystyle \int \sqrt{x^2 + 2x}\ dx $
$ \displaystyle \int \frac{x^2 + 1}{(x^2 - 2x +2)^2}\ dx $
$ \displaystyle \int x \sqrt{1 - x^4}\ dx $
$ \displaystyle \int_0^{\frac{\pi}{2}} \frac{\cos t}{\sqrt{1 + \sin^2 t}}\ dt $
(a) Use trigonometric substitution to show that$$ \displaystyle \int \frac{dx}{\sqrt{x^2 + a^2}} = \ln (x + \sqrt{x^2 + a^2}) + C $$(b) Use the hyperbolic substitution $ x = a \sinh t $ to show that$$ \displaystyle \int \frac{dx}{\sqrt{x^2 + a^2}} = \sinh^{-1} \left (\frac{x}{a} \right) + C $$These formulas are connected by Formula 3.11.3.
Evaluate$$ \displaystyle \int \frac{x^2}{(x^2 + a^2)^{\frac{3}{2}}}\ dx $$(a) by trigonometric substitution.(b) by the hyperbolic substitution $ x = a \sinh t $.
Find the average value of $ f(x) = \frac{\sqrt{x^2 -1}}{x} $ , $ 1 \le x \le 7 $.
Find the area of the region bounded by the hyperbola $ 9x^2 - 4y^2 = 36 $ and the line $ x = 3 $.
Prove the formula $ A = \frac{1}{2} r^2 \theta $ for the area of a sector of a circle with radius $ r $ and central angle $ \theta $. [Hint: Assume $ 0 < 0 < \frac{\pi}{2} $ and place the center of the circle at theorigin so it has the equation $ x^2 + y^2 = r^2 $.Then $ A $ is the sum of the area of the triangle $ POQ $ and the area of the region $ PQR $ in the figure.]
Evaluate the integral$$ \displaystyle \int \frac{dx}{x^4 \sqrt{x^2 - 2}} $$Graph the integrand and its indefinite integral on the same screen and check that your answer is reasonable.
Find the volume of the solid obtained by rotating about the x-axis the region enclosed by the curves $ y = \frac{9}{(x^2 + 9)} $ , $ y = 0 $ , $ x = 0 $ , and $ x = 3 $.
Find the volume of the solid obtained by rotating about the line $ x = 1 $ the region under the curve $ y = x \sqrt{1 - x^2} $ , $ 0 \le x \le 1 $.
(a) Use trigonometric substitution to verify that $$ \displaystyle \int_0^x \sqrt{a^2 - t^2}\ dt = \frac{1}{2} a^2 \sin^{-1} \left (\frac{x}{a} \right) + \frac{1}{2} x \sqrt{a^2 - x^2} $$(b) Use the figure to give trigonometric interpretations of both terms on the right side of the equation in part (a).
The parabola $ y = \frac{1}{2} x^2 $ divides the disk $ x^2 + y^2 \le 8 $ into two parts. Find the areas of both parts.
A torus is generated by rotating the circle $ x^2 + (y - R)^2 = r^2 $ about the x-axis. Find the volume enclosed by the torus.
A charged rod of length $ L $ produces an electric field at point $ P(a, b) $ given by$$ E(P) = \int_{-a}^{L - a} \frac{\lambda b}{4 \pi \varepsilon_0 (x^2 + b^2)^{\frac{3}{2}}}\ dx $$where $ \lambda $ is the charge density per unit length on the rod and $ \varepsilon_0 $ is the free space permittivity (see the figure). Evaluate the integral to determine an expression for the electric field $ E(P) $.
Find the area of the crescent-shaped region (called a $ lune $) bounded by arcs of circles with radii $ r $ and $ R $. (See the figure.)
A water storage tank has the shape of a cylinder with diameter 10 ft. It is mounted so that the circular cross-sections are vertical. If the depth of the water is 7 ft, what percentage of the total capacity is being used?
