Find the integral: ∫ 3 2 t 2 sin − 1 ⁡ ( t ) d t.

Question

Find the integral:   ∫      3    2        ﻿        t          2            sin          −      1        ⁡  (  t  )  d  t.

Taniya Wood · Accepted Answer

This integral could not be completed using integration by parts. Mathway will use another method.

Simplify.

$\int \frac{3}{2} t^{2} - \sin (t) d t$

Split the single integral into multiple integrals.

$\int \frac{3}{2} t^{2} d t + \int - \sin (t) d t$

Since $\frac{3}{2}$ is constant with respect to $t$ , move $\frac{3}{2}$ out of the integral.

$\frac{3}{2} \int t^{2} d t + \int - \sin (t) d t$

By the Power Rule, the integral of $t^{2}$ with respect to $t$ is $\frac{1}{3} t^{3}$ .

$\frac{3}{2} (\frac{1}{3} t^{3} + C) + \int - \sin (t) d t$

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

$\frac{3}{2} (\frac{1}{3} t^{3} + C) - \int \sin (t) d t$

The integral of $\sin (t)$ with respect to $t$ is $- \cos (t)$ .

$\frac{3}{2} (\frac{1}{3} t^{3} + C) - (- \cos (t) + C)$

Simplify.

$\frac{1}{2} t^{3} + \cos (t) + C$

Find the integral: 3 2  </mrow> t 2 </mrow>