Compute e t </mrow> &#xFEFF; </mrow> <msqrt> 9

lasquiyas5loaa

lasquiyas5loaa

Answered question

2022-04-10

Compute e t  9 e 2 t + 4 d t.

Answer & Explanation

Cara Cannon

Cara Cannon

Beginner2022-04-11Added 14 answers

Take the integral:
 et9e2t+4dt

For the integrand et9e2t+4, substitute u=et and du=etdt:
=9u2+4du

For the integrand 9u2+4, substitute u=2tans3 and du=2sec2s3ds. Then 9u2+4=4tan2s+4=2secs and s=tan-13u2:
=232sec3sds

Factor out constants:
=43sec3sds

Use the reduction formula, secmsds=sinssecm-1sm-1+m-2m-1sec-2+msds, where m = 3:
=23tanssecs+23secsds

The integral of secs is logtans+secs:
=23tanssecs+23logtans+secs+С

Substitute back for s=tan-13u2:
=23tantan-13u2sectan-13u2+23logtantan-13u2+sectan-13u2+С

Simplify using sectan-1z=z2+1 and tantan-1z=z:
=9u24+1u+23log129u2+4+3u+С

Substitute back for u=et:
=et9e2t4+1+23log129e2t+4+3et+С

Factor the answer a different way:
=163et9e2t+4+4log129e2t+4+3et+С

Which is equivalent for restricted t values to:
Answer:
=163et9e2t+4+4sinh-13et2+С

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