King Saud University Math 106Science and Medical Studies Section for girls First Term 1432-33HCollege of Science Second Midterm ExamDepartment of Mathematics 105 Minutes

Name: Student No.:

Section No.: Staff member name:

Question No I II III IV TotalMark

Question IEvaluate the following integrals:

1) ∫0

sinh3 1√1+x2

dx

2) ∫0

∞ x(1+x2)2

dx

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3) ∫ (lnx )2dx

4) ∫ 3x2−4 x+8

dx

5) ∫cos3x √sinx dx

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Question II

1) Use the substitution u=x2+1 to prove that

∫ 2x3+ x

dx=∫ 1u (u−1 )

du.

2) Compute ∫ 1u(u−1)

du.

3) Conclude ∫ 2x3+ x

dx.

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Question III

Use an appropriate trigonometric substitution to show that

∫√1+4 x2dx=12∫ sec3θdθ.

Question IV1) Evaluate f ' (0) for f ( x )=tanh−15 x.

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2) Determine whether the improper integral ∫0

π

sec2 xdx converges or diverges.

Good Luck

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