Integrate 5 4 t sin − 1 </mrow> &#x2061;<!-- ⁡

vilitatelp014

vilitatelp014

Answered question

2022-05-13

Integrate 5 4 t sin 1 ( t ) with respect to t.

Answer & Explanation

Braxton Gallagher

Braxton Gallagher

Beginner2022-05-14Added 21 answers

Simplify.

5tsin-1(t)4dt

Since 54 is constant with respect to t, move 54 out of the integral.

54tsin-1(t)dt

Integrate by parts using the formula udv=uv-vdu, where u=sin-1(t) and dv=t.

54(sin-1(t)(12t2)-12t211-t2dt)

Simplify.

54(sin-1(t)t22-t221-t2dt)

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

54(sin-1(t)t22-(12t21-t2dt))

Let t=sin(t1), where -π2t1π2. Then dt=cos(t1)dt1. Note that since -π2t1π2cos(t1) is positive.

54(sin-1(t)t22-12sin2(t1)1-sin2(t1)cos(t1)dt1)

Simplify terms.

54(sin-1(t)t22-12sin2(t1)dt1)

Use the half-angle formula to rewrite sin2(t1) as 1-cos(2t1)2.

54(sin-1(t)t22-121-cos(2t1)2dt1)

Since 12 is constant with respect to t1, move 12 out of the integral.

54(sin-1(t)t22-12(121-cos(2t1)dt1))

Simplify.

54(sin-1(t)t22-141-cos(2t1)dt1)

Split the single integral into multiple integrals.

54(sin-1(t)t22-14(dt1+-cos(2t1)dt1))

Apply the constant rule.

54(sin-1(t)t22-14(t1+C+-cos(2t1)dt1))

Since -1 is constant with respect to t1, move -1 out of the integral.

54(sin-1(t)t22-14(t1+C-cos(2t1)dt1))

Let u=2t1. Then du=2dt1, so 12du=dt1. Rewrite using u and du.

54(sin-1(t)t22-14(t1+C-cos(u)12du))

Combine cos(u) and 12.

54(sin-1(t)t22-14(t1+C-cos(u)2du))

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

54(sin-1(t)t22-14(t1+C-(12cos(u)du)))

The integral of cos(u) with respect to u is sin(u).

54(sin-1(t)t22-14(t1+C-12(sin(u)+C)))

Simplify.

54(sin-1(t)t22-t14+sin(u)8)+C

Substitute back in for each integration substitution variable.

54(sin-1(t)t22-sin-1(t)4+sin(2sin-1(t))8)+C

Simplify.

5t2sin-1(t)8-5sin-1(t)16+5sin(2sin-1(t))32+C

Reorder terms.

58t2sin-1(t)-516sin-1(t)+532sin(2sin-1(t))+C

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