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Defensorentx9

Defensorentx9

Answered question

2022-05-11

Integrate 7 2  tan ( 2 x ) sec ( 2 x ) ln ( sec ( 2 x ) ) with respect to x.

Answer & Explanation

Leroy Lowery

Leroy Lowery

Beginner2022-05-12Added 22 answers

Simplify.

7sec(2x)tan(2x)ln(sec(2x))2dx

Since 72 is constant with respect to x, move 72 out of the integral.

72sec(2x)tan(2x)ln(sec(2x))dx

Let u2=sec(2x). Then du2=2sec(2x)tan(2x)dx, so 12du2=sec(2x)tan(2x)dx. Rewrite using u2 and du2.

72ln(u2)12du2

Combine ln(u2) and 12.

72ln(u2)2du2

Since 12 is constant with respect to u2, move 12 out of the integral.

72(12ln(u2)du2)

Simplify.

74ln(u2)du2

Integrate by parts using the formula wdv=wv-vdw, where w=ln(u2) and dv=1.

74(ln(u2)u2-u21u2du2)

Simplify.

74(ln(u2)u2-du2)

Apply the constant rule.

74(ln(u2)u2-(u2+C))

Simplify.

74(ln(u2)u2-u2)+C

Replace all occurrences of u2 with sec(2x).

74(ln (sec(2x))sec(2x)-sec(2x))+C

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