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Chapter 6 | Applications of Integration
6.9 EXERCISES 377. [T] Find expressions for cosh x +sinh x and cosh x −sinh x . Use a calculator to graph these functions and ensure your expression is correct. 378. From the definitions of cosh( x ) and sinh( x ), find their antiderivatives. 379. Show that cosh( x ) and sinh( x ) satisfy y ″= y . tanh( x )′ = sech 2 ( x ). 381. Derive cosh 2 ( x )+sinh 2 ( x ) =cosh(2 x ) from the definition. 382. Take the derivative of the previous expression to find an expression for sinh(2 x ). 380. Use the quotient rule to verify that
For the following exercises, find the antiderivatives for the given functions.
395. cosh(2 x +1) 396. tanh(3 x +2)
397. x cosh ⎛
⎞ ⎠
⎝ x 2
398. 3 x 3 tanh ⎛
⎞ ⎠
⎝ x 4
399. cosh 2 ( x )sinh( x ) 400. tanh 2 ( x )sech 2 ( x )
sinh( x ) 1+cosh( x )
401.
383.
Prove
402. coth( x ) 403. cosh( x )+sinh( x ) 404. ⎛ ⎝ cosh( x )+sinh( x )
sinh( x + y ) = sinh( x )cosh( y )+cosh( x )sinh( y )
by
changing the expression to exponentials. 384. Take the derivative of the previous expression to find an expression for cosh( x + y ). For the following exercises, find the derivatives of the given functions and graph along with the function to ensure
n
⎞ ⎠
For the following exercises, find the derivatives for the functions. 405. tanh −1 (4 x ) 406. sinh −1 ⎛ ⎝ x 2 ⎞ ⎠
your answer is correct. 385. [T] cosh(3 x +1)
386. [T] sinh ⎛
⎞ ⎠
⎝ x 2
407. sinh −1 ⎛ 408. cosh −1 ⎛ ⎞ ⎠ 409. tanh −1 (cos( x )) ⎝ cosh( x ) ⎞ ⎠ ⎝ x 3
1 cosh( x )
387. [T]
388. [T] sinh ⎛
⎝ ln( x ) ⎞ ⎠
389. [T] cosh 2 ( x )+sinh 2 ( x ) 390. [T] cosh 2 ( x )−sinh 2 ( x )
x )
410. e sinh −1 (
411. ln ⎛
⎝ tanh −1 ( x ) ⎞ ⎠
391. [T] tanh ⎛
⎞ ⎠
⎝ x 2 +1
For the following exercises, find the antiderivatives for the functions. 412. ∫ dx 4− x 2 413. ∫ dx a 2 − x 2
x ) 1−tanh( x )
392. [T] 1+tanh(
393. [T] sinh 6 ( x ) 394. [T] ln ⎛
⎝ sech( x )+tanh( x ) ⎞ ⎠
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