AB Chapter 7 Review

Integration by Parts

200

Evaluate:  

int lnx/(x^2)dx

-lnx/x - 1/x + C

300

Evaluate:  

inte^(2x)sinxdx

2/5e^(2x)sinx - 1/5e^(2x)cosx + C

400

Evaluate:  

int_0^1tan^-1xdx

pi/4 - 1/2ln2

500

Evaluate: 

int(xe^(2x))/(1 + 2x)^2dx

-1/2(xe^(2x))/(1 + 2x) + 1/4e^(2x) + C

(e^(2x))/(4(1 + 2x)) + C

U-Sub

200

Evaluate:  

int2^t/(2^t + 3)dt

1/ln2ln(2^x + 3) + C

300

Evaluate: 

intdt/(cos^2tsqrt(1 + tant)

2sqrt(1 + tanx) + C

400

Evaluate:  

intlnx/(xsqrt(1 + (lnx)^2))dx

sqrt(1 + (lnx)^2) + C

500

Evaluate:  

intx^3sqrt(x^2 + 1)dx

1/5(x^2 + 1)^(5/2) - 1/3(x^2 + 1)^(3/2) + C

Trigonometric Integrals

200

Evaluate: 

intcot^2xdx

-x - cotx + C

300

Evaluate: 

intsec^5xtan^5xdx

1/9sec^9x - 2/7sec^7x + 1/5sec^5x + C

400

Evaluate: 

intsin^7xdx

-cosx + cos^3x - 3/5cos^5x + 1/7cos^7 x + C

500

Evaluate: 

intcos^6xdx

5/16x + 1/4sin(2x) + 3/64sin(4x) - 1/48sin^3(2x) + C

Other Strategies

200

Evaluate: 

intdx/(1- sinx)

tanx + secx + C

300

Evaluate: 

intsqrt(1 - x)/(sqrt(1 + x))dx

sin^(-1)x + sqrt(1 - x^2) + C

400

Evaluate:  

int (x - 3)/(2 + x^2)dx

1/2ln(x^2 + 2) - 3/sqrt2tan^(-1)(x/sqrt2) + C

500

Evaluate:  

int(x^3 + 3x^2 + 8x + 19)/(x^2 + 5)dx

1/2x^2 + 3x + 3/2ln(x^2 + 5) + 4/sqrt5tan^(-1)(x/sqrt5) + C

Misc.

200

Evaluate:  

int1/(x(lnx)^3)dx

-1/(2(lnx)^2) + C

300

Evaluate: 

int x^2cosxdx

x^2sinx + 2xcosx - 2sinx + C

400

Approximate the area under the curve using a trapezoidal approximation with 6 evenly spaced subintervals. Provide an answer that is a simplified, improper fraction.

int_-4^5x^2dx

531/8

500

Evaluate:  

intx^2sqrt(x - 1) dx

2/7(x - 1)^(7/2) + 4/5(x - 1)^(5/2) + 2/3(x - 1)^(3/2) + C