AP Calculus BC Unit 6: Integration and Accumulation of Change — Free with Answer Explanations

Question 1

Evaluate $\int (3 x^{2} - 4 x + 5) d x$ .

A.

6 x - 4 + C

B.

x^{3} - 2 x^{2} + 5 x

C.

x^{3} - 2 x^{2} + 5 x + C

D.

x^{3} - 4 x^{2} + 5 x + C

Question 2

Evaluate $\int \frac{x}{4 - x ^{2}} d x$ .

A.

arcsin (\frac{x}{2}) + C

B.

- 4 - x^{2} + C

C.

4 - x^{2} + C

D.

- \frac{1}{2} 4 - x^{2} + C

Question 3

Evaluate $\int \frac{2 x}{x ^{2} + 3} d x$ .

A.

ln ∣ x^{2} + 3∣ + C

B.

2 ln ∣ x^{2} + 3∣ + C

C.

\frac{- 2 x}{( x ^{2} + 3 ) ^{2}} + C

D.

\frac{1}{2} ln ∣ x^{2} + 3∣ + C

Question 4

Evaluate $\int x cos x d x$ using integration by parts.

A.

x sin x + cos x + C

B.

x sin x - cos x + C

C.

cos x - x sin x + C

D.

x cos x + sin x + C

Question 5

Evaluate $\int_{0}^{π} sin^{2} x d x$ .

A.

0

B.

\frac{π}{2}

C.

π

D.

1

Question 6

Evaluate $\int x e^{x^{2}} d x$ .

A.

\frac{1}{2} e^{x^{2}} + C

B.

e^{x^{2}} + C

C.

x^{2} e^{x^{2}} /2 + C

D.

2 x e^{x^{2}} + C

Question 7

Evaluate $\int x^{2} e^{x} d x$ .

A.

e^{x} (x^{2} - 2 x + 2) + C

B.

e^{x} (x^{2} + 2 x + 2) + C

C.

e^{x} (x^{2} - 2 x) + C

D.

x^{2} e^{x} - e^{x} + C

Question 8

Using the Left Riemann Sum with $n = 4$ equal subintervals, approximate $\int_{0}^{2} x^{2} d x$ .

A.

\frac{7}{4}

B.

\frac{15}{4}

C.

\frac{8}{3}

D.

\frac{11}{4}

Question 9

Evaluate $\int sin x cos x d x$ .

A.

- cos^{2} x + C

B.

\frac{sin ^{2} x}{2} + C

C.

cos^{2} x + C

D.

sin^{2} x cos x + C

Question 10

Evaluate $\int_{1}^{4} x d x$ .

A.

\frac{14}{3}

B.

\frac{4}{3}

C.

\frac{2}{3} (8 - 1) = \frac{14}{3}

... wait:

3

D.

\frac{7}{3}

Question 11

If $F (x) = \int_{0}^{x} cos (t^{2}) d t$ , find $F^{'} (x)$ by the Fundamental Theorem of Calculus Part 1.

A.

cos (x)

B.

cos (x^{2})

C.

2 x cos (x^{2})

D.

sin (x^{2})

Question 12

If $G (x) = \int_{1}^{x^{3}} ln t d t$ , find $G^{'} (x)$ .

A.

ln (x^{3})

B.

3 x^{2} ln (x^{3})

C.

\frac{3 x ^{2}}{x ^{3}}

D.

3 ln x