Stewart 8th Section 7.3: Trigonometric Substitution

1 Exercise - Trigonometric Substitution (Guided) · Level 2

Evaluate the integral using the indicated trigonometric substitution. Sketch and label the associated right triangle: \(\int \dfrac{d x}{x^2 \sqrt{4 - x^2}}\), \(x = 2 \sin \theta\).

2 Exercise - Trigonometric Substitution (Guided) · Level 2

Evaluate the integral using the indicated trigonometric substitution. Sketch and label the associated right triangle: \(\int \dfrac{x^3}{\sqrt{x^2 + 4}} d x\), \(x = 2 \tan \theta\).

3 Exercise - Trigonometric Substitution (Guided) · Level 2

Evaluate the integral using the indicated trigonometric substitution. Sketch and label the associated right triangle: \(\int \dfrac{\sqrt{x^2 - 4}}{x} d x\), \(x = 2 \sec \theta\).

4 Exercise - Trigonometric Substitution · Level 2

Evaluate the integral: \(\int \dfrac{x^2}{\sqrt{9 - x^2}} d x\).

5 Exercise - Trigonometric Substitution · Level 2

Evaluate the integral: \(\int \dfrac{\sqrt{x^2 - 1}}{x^4} d x\).

6 Exercise - Trigonometric Substitution · Level 2

Evaluate the integral: \(\displaystyle\int_{0}^{3} \dfrac{x}{\sqrt{36 - x^2}} d x\).

7 Exercise - Trigonometric Substitution · Level 2

Evaluate the integral: \(\displaystyle\int_{0}^{a} \dfrac{d x}{(a^2 + x^2)^{\dfrac{3}{2}}}\), \(a > 0\).

8 Exercise - Trigonometric Substitution · Level 2

Evaluate the integral: \(\int \dfrac{d t}{t^2 \sqrt{t^2 - 16}}\).

9 Exercise - Trigonometric Substitution · Level 2

Evaluate the integral: \(\displaystyle\int_{2}^{3} \dfrac{d x}{(x^2 - 1)^{\dfrac{3}{2}}}\).

10 Exercise - Trigonometric Substitution · Level 2

Evaluate the integral: \(\displaystyle\int_{0}^{\dfrac{2}{3}} \sqrt{4 - 9 x^2} d x\).

11 Exercise - Trigonometric Substitution · Level 2

Evaluate the integral: \(\displaystyle\int_{0}^{\dfrac{1}{2}} x \sqrt{1 - 4 x^2} d x\).

12 Exercise - Trigonometric Substitution · Level 2

Evaluate the integral: \(\displaystyle\int_{0}^{2} \dfrac{d t}{\sqrt{4 + t^2}}\).

13 Exercise - Trigonometric Substitution · Level 2

Evaluate the integral: \(\int \dfrac{\sqrt{x^2 - 9}}{x^3} d x\).

14 Exercise - Trigonometric Substitution · Level 2

Evaluate the integral: \(\displaystyle\int_{0}^{1} \dfrac{d x}{(x^2 + 1)^2}\).

15 Exercise - Trigonometric Substitution · Level 3

Evaluate the integral: \(\displaystyle\int_{0}^{a} x^2 \sqrt{a^2 - x^2} d x\).

16 Exercise - Trigonometric Substitution · Level 3

Evaluate the integral: \(\displaystyle\int_{\dfrac{\sqrt{2}}{3}}^{\dfrac{2}{3}} \dfrac{d x}{x^5 \sqrt{9 x^2 - 1}}\).

17 Exercise - Trigonometric Substitution · Level 2

Evaluate the integral: \(\int \dfrac{x}{\sqrt{x^2 - 7}} d x\).

18 Exercise - Trigonometric Substitution · Level 3

Evaluate the integral: \(\int \dfrac{d x}{[(a x)^2 - b^2]^{\dfrac{3}{2}}}\).

19 Exercise - Trigonometric Substitution · Level 2

Evaluate the integral: \(\int \dfrac{\sqrt{1 + x^2}}{x} d x\).

20 Exercise - Trigonometric Substitution · Level 1

Evaluate the integral: \(\int \dfrac{x}{\sqrt{1 + x^2}} d x\).

21 Exercise - Trigonometric Substitution · Level 2

Evaluate the integral: \(\displaystyle\int_{0}^{0}.6 \dfrac{x^2}{\sqrt{9 - 25 x^2}} d x\).

22 Exercise - Trigonometric Substitution · Level 3

Evaluate the integral: \(\displaystyle\int_{0}^{1} \sqrt{x^2 + 1} d x\).

23 Exercise - Complete the Square · Level 2

Evaluate the integral: \(\int \dfrac{d x}{\sqrt{x^2 + 2 x + 5}}\).

24 Exercise - Complete the Square · Level 2

Evaluate the integral: \(\displaystyle\int_{0}^{1} \sqrt{x - x^2} d x\).

25 Exercise - Complete the Square · Level 3

Evaluate the integral: \(\int x^2 \sqrt{3 + 2 x - x^2} d x\).

26 Exercise - Complete the Square · Level 3

Evaluate the integral: \(\int \dfrac{x^2}{(3 + 4 x - 4 x^2)^{\dfrac{3}{2}}} d x\).

27 Exercise - Complete the Square · Level 2

Evaluate the integral: \(\int \sqrt{x^2 + 2 x} d x\).

28 Exercise - Complete the Square · Level 3

Evaluate the integral: \(\int \dfrac{x^2 + 1}{(x^2 - 2 x + 2)^2} d x\).

29 Exercise - Trigonometric Substitution · Level 2

Evaluate the integral: \(\int x \sqrt{1 - x^4} d x\).

30 Exercise - Trigonometric Substitution · Level 2

Evaluate the integral: \(\displaystyle\int_{0}^{\dfrac{\pi}{2}} \dfrac{\cos t}{\sqrt{1 + \sin^2 t}} d t\).

31 Exercise - Proof · Level 3

(a) Use trigonometric substitution to show that \(\int \dfrac{d x}{\sqrt{x^2 + a^2}} = \ln(x + \sqrt{x^2 + a^2}) + C\). (b) Use the hyperbolic substitution \(x = a \sinh t\) to show that \(\int \dfrac{d x}{\sqrt{x^2 + a^2}} = \sinh^{-1}\left(\dfrac{x}{a}\right) + C\). These formulas are connected by Formula 3.11.3.

32 Exercise - Two Methods · Level 3

Evaluate \(\int \dfrac{x^2}{(x^2 + a^2)^{\dfrac{3}{2}}} d x\): (a) by trigonometric substitution. (b) by the hyperbolic substitution \(x = a \sinh t\).

33 Exercise - Average Value · Level 2

Find the average value of \(f(x) = \dfrac{\sqrt{x^2 - 1}}{x}\), \(1 \leq x \leq 7\).

34 Exercise - Area of Region · Level 3

Find the area of the region bounded by the hyperbola \(9 x^2 - 4 y^2 = 36\) and the line \(x = 3\).

35 Exercise - Sector Area Proof · Level 3

Prove the formula \(A = \dfrac{1}{2} r^2 \theta\) for the area of a sector of a circle with radius \(r\) and central angle \(\theta\). [Hint: Assume \(0 < \theta < \dfrac{\pi}{2}\) and place the center of the circle at the origin so it has the equation \(x^2 + y^2 = r^2\). Then \(A\) is the sum of the area of the triangle \(P O Q\) and the area of the region \(P Q R\) in the figure.]

36 Exercise - Graphical Verification · Level 3

Evaluate the integral \(\int \dfrac{d x}{x^4 \sqrt{x^2 - 2}}\). Graph the integrand and its indefinite integral on the same screen and check that your answer is reasonable.

37 Exercise - Volume of Revolution · Level 3

Find the volume of the solid obtained by rotating about the \(x\)-axis the region enclosed by the curves \(y = 9/(x^2 + 9)\), \(y = 0\), \(x = 0\), and \(x = 3\).

38 Exercise - Volume of Revolution · Level 3

Find the volume of the solid obtained by rotating about the line \(x = 1\) the region under the curve \(y = x \sqrt{1 - x^2}\), \(0 \leq x \leq 1\).

39 Exercise - Geometric Interpretation · Level 3

(a) Use trigonometric substitution to verify that \(\displaystyle\int_{0}^{x} \sqrt{a^2 - t^2} d t = \dfrac{1}{2} a^2 \sin^{-1}\left(\dfrac{x}{a}\right) + \dfrac{1}{2} x \sqrt{a^2 - x^2}\). (b) Use the figure to give trigonometric interpretations of both terms on the right side of the equation in part (a).

40 Exercise - Area Division · Level 3

The parabola \(y = \dfrac{1}{2} x^2\) divides the disk \(x^2 + y^2 \leq 8\) into two parts. Find the areas of both parts.

41 Exercise - Torus Volume · Level 3

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.

42 Exercise - Electric Field Application · Level 3

A charged rod of length \(L\) produces an electric field at point \(P(a, b)\) given by \(E(P) = \displaystyle\int_{-a}^{L - a} \dfrac{\lambda b}{4 \pi \epsilon_0 (x^2 + b^2)^{\dfrac{3}{2}}} d x\) where \(\lambda\) is the charge density per unit length on the rod and \(\epsilon_0\) is the free space permittivity. Evaluate the integral to determine an expression for the electric field \(E(P)\).

43 Exercise - Lune Area · Level 3

Find the area of the crescent-shaped region (called a lune) bounded by arcs of circles with radii \(r\) and \(R\).

44 Exercise - Water Tank Application · Level 3

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?