Question

Carry out the following steps to show that $\int csch\ x\ dx = ln\left|tanh\frac{x}{2}\right| + C$. a) Change variables with the substitution $u = \frac{x}{2}$ to show that $\int csch\ x\ dx = \int \frac{2\ du}{sinh\ 2u}$. b) Use the identity $sinh\ 2u = 2sinh\ u\ cosh\ u$ to show that $\frac{2}{sinh\ 2u} = \frac{sech^2 u}{tanh\ u}$. c) Change variables again to determine $\int \frac{sech^2 u}{tanh\ u} du$. d) Express your answer in terms of x.

          Carry out the following steps to show that $\int csch\ x\ dx = ln\left|tanh\frac{x}{2}\right| + C$.

a) Change variables with the substitution $u = \frac{x}{2}$ to show that $\int csch\ x\ dx = \int \frac{2\ du}{sinh\ 2u}$.

b) Use the identity $sinh\ 2u = 2sinh\ u\ cosh\ u$ to show that $\frac{2}{sinh\ 2u} = \frac{sech^2 u}{tanh\ u}$.

c) Change variables again to determine $\int \frac{sech^2 u}{tanh\ u} du$.

d) Express your answer in terms of x.

Carry out the following steps to show that ∫ csch x dx = ln|tanh(x)/(2)| + C.

a) Change variables with the substitution u = (x)/(2) to show that ∫ csch x dx = ∫(2 du)/(sinh 2u).

b) Use the identity sinh 2u = 2sinh u cosh u to show that (2)/(sinh 2u) = (sech^2 u)/(tanh u).

c) Change variables again to determine ∫(sech^2 u)/(tanh u) du.

d) Express your answer in terms of x.

Added by Justin M.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Dharmendra Jain

09/17/2020

Step 1

Then $$du = \frac{1}{2} dx$$, so $$dx = 2du$$. Show more…

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Carry out the following steps to show that $$ \int csch x dx = ln|tanh \frac{x}{2}| + C.$$ a) Change variables with the substitution $$u = \frac{x}{2}$$ to show that $$ \int csch x dx = \int \frac{2du}{sinh 2u}.$$ b) Use the identity $$sinh 2u = 2 sinh u cosh u$$ to show that $$ \frac{2}{sinh 2u} = \frac{sech^2 u}{tanh u}.$$ c) Change variables again to determine $$ \int \frac{sech^2 u}{tanh u} du.$$ d) Express your answer in terms of x.