﻿ ﻿INTEGRATION BY SUBSTITUTION; CHANGE OF VARIABLE - Special Methods of Integration - The Calculus Primer

## The Calculus Primer (2011)

### Chapter 49. INTEGRATION BY SUBSTITUTION; CHANGE OF VARIABLE

13—5. Algebraic Substitution. Frequently an expression to be integrated can be transformed, by the suitable substitution of a new variable, into one of the fundamental standard forms. Some of the simpler kinds of such substitutions will now be illustrated.

EXAMPLE 1. Find  Solution.

Let  Substituting (3x − 2)½ for z: EXAMPLE 2. Find Solution.

Let z = (3x + l);

then dx = z2 dz, and (3x + 1) = z2. Substituting (3x + 1) = z: EXAMPLE 3. Find  Solution. Let x = z; then x = z6, dx = 6z5 dz, x = z3, and x = z2. Substituting for z, z2, and z3: EXAMPLE 4. Find Solution. Let z = ; then x = z2 + 3, and dx = 2z dz. Substituting  EXERCISE 13—3

Find the following:  Verify the following: 13—6. Trigonometric Substitutions. When the integrand contains expressions such as the integration may be performed by using the following trigonometric substitutions:

I.For we put v = a sin θ; the expression then becomes: II.For we put v = a tan θ; the expression then becomes: III.For we put v = a sec θ; the expression then becomes:  EXAMPLE 1. Find Solution. Let a2 = 9, a = 3; x = 3 sin z, dx = 3 cos z dz; Therefore, But, x = 3 sin z; from the triangle of reference we have z = arc sin , and cos hence EXAMPLE 2. Find   Solution. Let x = 2 tan z; dx = 2 sec2 z dz;  = log (sec z + tan z).

Therefore, from the triangle of reference, EXAMPLE 3. Find Solution. Let x = a sec z; dx = a sec z tan z dz; = a tan z.

Therefore,  But sec z = ; hence, from the triangle: EXERCISE 13—4

Verify the following: ﻿