Integration

Question 1. $~{5\over~6-7x}~$ choose correct answer below:

$~{-35\over~(6-7x)^2}~+~c~$

$~-{5\over7}~log(6-7x)~+~c~$

$~{5\over7}~log~(6-7x)~+~c~$

$~{-5\over~7(6-7x)}~+~c~~$

None of the above

Question 2. $~{1\over~\sqrt{x+6}}~$ choose correct answer below:

$~2(x+6)^{1/2}~+~c~$

$~-(1/2)(x+6)^{-3/2}~+~c~$

$~(1/3)(x+6)^{1/2}~+~c~$

$~2(x+6)^{-1/4}~+~c~$

None of the above

Question 3. $~2e^{(1-5x)}~$ choose correct answer below:

$~{2\over5}e^{1-5x}~+~c~~$

$~{2\over~1-5x}e^{1-5x}~+~c~~$

$~-10e^{(1-5x)}~$

$~-{2\over5}e^{1-5x}~+~c~$

None of the above

Question 4. $~3\cos~\left(4x-{\pi\over~2}\right)~$ choose correct answer below:

$~{3\over4}\sin(4x)~+~c~$

$~-12\sin\left(4x~-~{\pi\over~2}\right)~+~c~$

$~-{3\over4}\cos(4x)~+~c~$

$~-{3\over4}\sin\left(4x~-~{\pi\over~2}\right)~+~c~~$

None of the above

Question 5. $~\int_0^{2}~e^{2x}\,~dx~$ choose correct answer below:

$~(e^4~-~1)~$

$~{1\over2}(e^4~-~1)~$

$~{1\over2}(e^2~-~1)~~$

$~(e^2~-~1)~$

None of the above

Question 6. $~\int_0^{\pi~/2}\cos~x\,~{\sin}^3~x\,~dx~$ Hint: use the substitution u=sin(x).

Question 7. $~\int_e^{e^2}\log~x\,~dx~$
Hint: consider log(x) as log(x)*1 and use integration by parts. Choose correct answer below:

$~e^3~$

$~e^2-1~$

$~1-e~$

$~e^2~$

None of the above

Question 8. $~\int_{0}^{\pi~/4}\sqrt{1-\cos~2x}\,~dx~$
Hint: express cos(2x) in terms of sin(x). Choose correct answer below:

$~\sqrt2~-~1$

$~1-\sqrt2~$

$~\sqrt2/2~$

$~2\sqrt2~$

None of the above

Question 9. Find the area of the finite region which is enclosed by the curve $y=x^4$ , and the line $y=x/8$ as a fraction in lowest terms.

Question 10. Find the area of the finite region enclosed by the curve y=cos(x), the x-axis, and the lines $x=\pi~/6$ and $x=\pi/3$ . Choose correct answer below:

$~\sqrt3~-~1~$

$~\sqrt3~-~1/2~$

$~{1\over2}(\sqrt3~-~1)~$

$~{1\over2}\sqrt3~-~1~$

None of the above

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Mathematics Institute

Integration

Score 8 or more out of 10 to pass.

Section A: Integrate the following expressions with respect to x.

Section B: Evaluate the following definite integrals.

Section C:

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