Trigonometric substitutions are often useful for integrals containing factors of the form (a2-x2)n (x2+a2)n or (x2-a2)n. The exact substitution used depends on the form of the integral: (a2-x2)n uses x = a sin θ, (x2+a2)n uses x = a tan θ, and (x2-a2)n uses x = a sec θ. These substitutions replace the integral variable with a trigonometric function of a new variable and allow the integral to be evaluated.

1. Algebraic Substitution
Example 1
Let
answer
Example 2
Let
Example 3

2. Let
x=asin x=atan x=asec
dx=acos dx=asec2 dx=asec tan
a2−x2=acos x2+a2=asec x2−a2=atan
Example 1
First see whether the quadratic fits the pattern by completing the square.
x2 + 12x + 45 = (x + 6)2 + 9.
It does. Set x = 3tan w − 6. then dx = 3sec2 w and w = arctan( ).
Example 2
First see whether the quadratic fits the pattern by completing the square.
9 + 8x − x2 = 25 − (x − 4)2.
It does. Substitute x = 5sin w + 4 , then dx = 5cos w and w = arcsin( ).
answer
Example 1 Evaluate the following integral.
Trigonometric substitutions are often useful for integrals containing factors
of the form (a2−x2)n (x2+a2)n or(x2−a2)n The exact substitution used
depends on the form of the integral:
(a2−x2)n (x2+a2)n (x2−a2)n

3. Problem 310
Show that the hollow circular shaft whose inner diameter is half the outer
diameter has a torsional strength equal to 15/16 of that of a solid shaft of the
same outside diameter.
Solution 310
Hollow circular shaft:
Solid circular shaft:
ok!
Problem 311
An aluminum shaft with a constant diameter of 50 mm is loaded by torques
applied to gears attached to it as shown in Fig. P-311. Using G = 28 GPa,
determine the relative angle of twist of gear D relative to gear A.

4. Problem 311
answer
Problem 314
The steel shaft shown in Fig. P-314 rotates at 4 Hz with 35 kW taken off at A,
Rotation of D relative to A:
20 kW removed at B, and 55 kW applied at C. Using G = 83 GPa, find the
maximum shearing stress and the angle of rotation of gear A relative to gear
C.
answer
Problem 312
Solution 314
A flexible shaft consists of a 0.20-in-diameter steel wire encased in a
stationary tube that fits closely enough to impose a frictional torque of 0.50
lb·in/in. Determine the maximum length of the shaft if the shearing stress is
not to exceed 20 ksi. What will be the angular deformation of one end
relative to the other end? G = 12 × 106 psi.
Solution 312
Relative to C:
If θ = dθ, T = 0.5L and L = dL

5. For AB
∴ answer
For BC
For CD
answer Use d = 69.6 mm answer
Problem 315
A 5-m steel shaft rotating at 2 Hz has 70 kW applied at a gear that is 2 m from Part (b)
the left end where 20 kW are removed. At the right end, 30 kW are removed
and another 20 kW leaves the shaft at 1.5 m from the right end. (a) Find the
uniform shaft diameter so that the shearing stress will not exceed 60 MPa.
(b) If a uniform shaft diameter of 100 mm is specified, determine the angle
by which one end of the shaft lags behind the other end. Use G = 83 GPa.
Solution 315
answer
Part (a)