Review the following concepts if needed:
 Antiderivatives Help
 The Indefinite Integral Help
 Area Under a Curve Help
 The Fundamental Theorem of Calculus Help
 Signed Area Help
 The Area Between Two Curves Help
 Rules of Integration Help
The Integral Practice Test

Calculate each of the following antiderivatives:
(a) Antiderivative of x ^{2} − sin x
(b) Antiderivative of e ^{3 x} + x ^{4} − 2
(c) Antiderivative of 3 t ^{2} + (In t/t )
(d) Antiderivative of tan x − cos x + sin 3 x
(e) Antiderivative of cos 3 x + sin 4 x + 1
(f) Antiderivative of (cos x ) · e ^{sin x}

Calculate each of the following indefinite integrals:
(a) ∫ x sin x ^{2} dx
(b) ∫ (3/ x ) In x ^{2} dx
(c) ∫ sin x · cos x dx
(d) ∫ tan x · In cos x dx
(e) ∫ sec ^{2} x · e ^{tan x} dx
(f) ∫ (2 x + 1) · ( x ^{2} + x + 7) ^{43} dx

Use Riemann sums to calculate each of the following integrals:

Use the Fundamental Theorem of Calculus to evaluate each of the following integrals:

Calculate the area under the given function and above the x axis over the indicated interval.
(a) f ( x ) = x ^{2} + x + 6 [2, 5]
(b) g ( x ) = sin x cos x [0, /4]
(c) h ( x ) = xe ^{x} ^{2} [1, 2]
(d) k ( x ) = In x/x [1, e ]

Draw a careful sketch of each function on the given interval, indicating subintervals where area is positive and area is negative.
(a) f ( x ) = x ^{3} + 3 x [−2, 2]
(b) g ( x ) = sin 3 x cos 3 x [−2 , 2 ]
(c) h ( x ) = In x/x [1/2, e ]
(d) m ( x ) = x ^{3} e ^{x} ^{4} [−3, 3]

For each function in Exercise 6, calculate the positive area between the graph of the given function and the x axis over the indicated interval.

In each part of Exercise 6, calculate the signed area between the graph of the given function and the x axis over the indicated interval.

Calculate the area between the two given curves over the indicated interval.

Calculate the area enclosed by the two given curves.
Answers

(a) F ( x ) = x ^{3} /3 + cos x + C
(b) F ( x ) = e ^{3 x} /3 + x ^{5} /5 − 2 x + C
(c) F ( t ) = t ^{3} + [ln t ] ^{2} /2
(d) F ( x ) = −ln(cos x ) − sin x − [cos 3 x ]/3 + C
(e) F ( x ) = [sin 3 x ]/3 − [cos4 x ]/4 + x + C
(f) F ( x ) = e ^{sin x} + C

(a)

(a) We have
(b) We have







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