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Chapter 5: Q. 6 (page 494)

If fand uare functions such that role="math" localid="1652080830652" f'(u(x))u'(x)is role="math" localid="1652080865441" , then we can use integration by substitution to solve

role="math" localid="1652080904444" 鈭�f'(u(x))u'(x)dx=

Short Answer

Expert verified

鈭�f'(u(x))u'(x)dx=fux+C

Step by step solution

01

Step 1. 

We know that in any integral 鈭�fxdx, fxis integrand.

And If localid="1652080706482" 鈭�f'(u(x))u'(x)dxis needed to be integrated by substitution then

We substitute localid="1652080668665" u(x)=v

hence,localid="1652080598731" dvdx=u'xdv=u'x.dx

Thus the given integral become

localid="1652080932120" 鈭�f'(u(x))u'(x)dx=鈭�f'(v)dv鈬�鈭�f'(u(x))u'(x)dx=f(v)+C

Substituting back

We get,鈭�f'(u(x))u'(x)dx=fux+C

02

Step 3. Final Answer

fux+CIf fand uare functions such that f'(u(x))u'(x)is integrand, then we can use integration by substitution to solve

鈭�f'(u(x))u'(x)dx=fux+C

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Most popular questions from this chapter

Solve given definite integrals.

鈭�04xx2+4dx

Solve 鈭�x+2x2+4x32dxthe following two ways:

(a) with the substitution u=x2+4x;

(b) by completing the square and then applying the trigonometric substitution x + 2 = 2 sec u.

Solve given definite integral.

鈭�45鈥�1xx2+9dx

Solve the integral:鈭�xcosxdx.

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) An integral with which we could reasonably apply trigonometric substitution with x=tan鈦�u.

(b) An integral with which we could reasonably apply trigonometric substitution with x=4sec鈦�u.

(c) An integral with which we could reasonably apply trigonometric substitution with x鈭�2=3sin鈦�u.
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