The Substitution Rule

Recall the Chain Rule:

(f(g(x)))′ = f′(g(x))g′(x)

What happens if we want to find ∫f′(g(x))g′(x)dx?

 

The Substitution Method: If F′(x) = f(x), then
∫f(u(x))u′ (x) dx = F(u(x)) + C.

Steps:
1.

2.

3.

Examples:
∫2x cos(x²) dx

 

 

 

 

Change of Variables Formula for Definite Integrals: If u(x) is Quadratic -formula/second-order-differential.html">differentiable on [a,b] and
f(x) is integrable on the range of u(x), then

49.

 

41.

 

Symmetry

 

48.

 

5.6 Integ ration by Parts

Recall the Product Rule :
(u(x)v(x))′ =

 

Integration By Parts Formula:∫u(x)v′(x) dx = u(x)v(x)−∫u′(x)v(x) dx
How to choose u and v′:
u:
v:
Examples:
4.∫xe^−x dx

 

∫ln x dx

 

∫x² cos x dx

 

∫e^−θcos2θ dθ

 

 

 

Reduction formulas
Prove the reduction formula:∫xⁿ cosx dx = xⁿ sinx−n∫xⁿ⁻¹ sin x dx:

5.7 Techniques of Integration: Trigonometric Integrals
Recall your trig identities!!!!
Examples:
Odd power of sin x:∫sin³ x dx

 

Odd power of sin x or cos x:∫sin⁴ x cos⁵ x dx

 

Even powers of sin x and cos x:∫sin² x cos⁴ x dx

 

8.tan² x sec⁴ x dx

 

5.7 Techniques of Integration: Trigonometric Substitution
Triangles!

 

Steps:
1.
2.
3.
Deciding on a substitution:

Square Root Form	Trigonometric Substitution in Integrand

Examples:
10.

 

14.

 

32.

 

5.7 Techniques of Integration: Partial Fractions
Partial Fraction Decomposition:

 

Linear Factors vs. Quadratic Factors :

 

Linear:
Repeated Linear:
Quadratic:
Repeated
Quadratic:

Example: Give the form of the partial fraction decomposition:

 

Solving for the constants:
Solve for the constants in the previous problem :

 

Integrating

18.

 

22.

 

Long Division

28.

 

5.9 Approximate Integration
Recall: Riemann Sums

 

Midpoint Rule: The Nth midpoint approximation tois
M_N = Δx(f(c₁) + f(c₂) +... + f(c_N)
whereand c_j is the midpoint of the jth interval [x_j−1,x_j ] or c_j = a+(j−.5)Δx.

 

Trapezoidal Rule: The Nth trapezoidal approximation tois
TN =Δx(y₀ + 2y₁ +...+ 2y_N−1 + y_N)
where Δx =and y_j = f(a + jΔx).

 

Simpson's Rule: Assume that N is even. LetΔx =and y_j = f(a + jΔx). The Nth
approximation toby Simpson's Rule is
S _N =Δx[y₀ + 4y₁ + 2y₂ +...+ 4y_N−3 + 2y_N−3 + 4y_N−1 + y_N]:

 

Example:
8. Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson's Rule to approx-
imatesin(x²) dx, with n = 4.

 

Error Bounds:
How accurate are these estimates?
Error Bound for T_N and M_N: Let K₂ be any number such that≤ K₂ for all x∈[a,b]. Then
E_T =Error(T_N)≤, E_N=Error(M_N)≤
Error Bound for S_N: Let K₄ be the number such that≤K₄ for all x∈[a,b]. Then
E_S=Error(S_N)≤

19. (a) Assume T₁₀ = 1.719713 and S10 = 1.718283 fore^x dx. Calculate the correspond -
ing errors E_T and E_S using the formulas given above.

 

(c) How large do we have to choose N so that the approximations T_N and S_N to the integral
in part (a) are accurate to within 0.00001?

 

5.10 Improper Integrals
Two main types of improper integrals:
Infinite Intervals
Three cases:

 

Convergent vs. Divergent:
Examples: De termine whether each of the following is convergent or divergent. Evaluate
those that are convergent.
5.

 

6.

 

12.(2−v⁴) dv

 

Example: For what values of p does the integralconverge?

 

**
Discontinuous Integrands
Three cases:

 

23.

 

24.

 

27.

 

Comparison Test: Assume that f(x)≥g(x)≥0 for x≥a.
•Ifconverges, thenconverges.
•Ifdiverges, thendiverges.

Examples: Use the Comparison Test to determine whether the integral is convergent or
divergent.
42.

 

44.