## The Calculus Primer (2011)

### Part XIII. Special Methods of Integration

### 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 (3*x −* 2)^{½} for *z:*

EXAMPLE 2. Find

*Solution.*

Let *z* = (3*x* + l)^{⅓};

then *dx* = *z*^{2} *dz*, and (3*x* + 1)^{⅔} = *z*^{2}.

Substituting (*3x +* 1)^{⅓} *= z*:

EXAMPLE 3. Find

*Solution.* Let *x*^{} = *z*; then *x* = *z*^{6}, *dx* = 6*z*^{5} *dz, x*^{⅓} = *z*^{3}, and *x*^{⅓} = *z*^{2}.

Substituting for *z, z*^{2}*,* and *z*^{3}:

EXAMPLE 4. Find

*Solution.* Let *z* = ; then *x* = *z*^{2} + 3, and *dx* = 2*z 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 *a*^{2} = 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 sec^{2} *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:*