Page 201 - Calculus Demystified

188 EXAMPLE 6.40 CHAPTER 6 Transcendental Functions
Evaluate the integral
sin x dx.
1 + cos x
2
SOLUTION
For clarity we set ϕ(x) = cos x, ϕ (x) =− sin x. The integral becomes

− ϕ (x) dx .
1 + ϕ (x)
2
By what we have just learned about Tan −1 , this last integral is equal to
−1
−Tan ϕ(x) + C.
Resubstituting ϕ(x) = cos x yields that
sin x
2 dx =−Tan −1 (cos x) + C.
1 + cos x
You Try It: Calculate x/(1 + x )dx.
4

EXAMPLE 6.41 TEAMFLY
Calculate the integral
3x 2
√ 1 − x 6 dx.

SOLUTION 3 2
For clarity we set ϕ(x) = x , ϕ (x) = 3x . The integral then becomes
ϕ (x) dx

1 − ϕ (x) .
2

We know that this last integral equals
Sin −1 ϕ(x) + C.

Resubstituting the formula for ϕ gives a final answer of
√ 3x 2 dx = Sin −1 (x ) + C.
3
1 − x 6

You Try It: Evaluate the integral

√ xdx .
1 − x 4

Team-Fly
®