Find the integral of 7 8 </mfrac> <mrow class="MJX-TeXAtom-ORD"> &#xFEFF; </mr

encamineu2cki

encamineu2cki

Answered question

2022-05-05

Find the integral of 7 8  e 2 t t 2 with respect to t.

Answer & Explanation

percolarse2rzd

percolarse2rzd

Beginner2022-05-06Added 17 answers

Simplify.

7e2tt28dt

Since 78 is constant with respect to t, move 78 out of the integral.

78e2tt2dt

Integrate by parts using the formula udv=uv-vdu, where u=t2 and dv=e2t.

78(t2(12e2t)-12e2t(2t)dt)

Simplify.

78(t2e2t2-e2ttdt)

Integrate by parts using the formula udv=uv-vdu, where u=t and dv=e2t.

78(t2e2t2-(t(12e2t)-12e2tdt))

Simplify.

78(t2e2t2-(te2t2-e2t2dt))

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

78(t2e2t2-(te2t2-(12e2tdt)))

Let u=2t. Then du=2dt, so 12du=dt. Rewrite using u and du.

78(t2e2t2-(te2t2-12eu12du))

Combine eu and 12.

78(t2e2t2-(te2t2-12eu2du))

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

78(t2e2t2-(te2t2-12(12eudu)))

Simplify.

78(t2e2t2-(te2t2-14eudu))

The integral of eu with respect to u is eu.

78(t2e2t2-(te2t2-14(eu+C)))

Rewrite 78(t2e2t2-(te2t2-14(eu+C))) as 78(t2e2t2-te2t2+eu4)+C.

78(t2e2t2-te2t2+eu4)+C

Replace all occurrences of u with 2t.

78(t2e2t2-te2t2+e2t4)+C

Reorder terms.

78(12t2e2t-12te2t+14e2t)+C

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