Integrate − t 3 2 + 5 2 t 2 + 2 t − 3 cos ⁡ ( t ) with respect to t.

Question

Integrate   −            t              3              2        ﻿    +      5          2              t                  2                      +  2  t  −  3  cos  ⁡  (  t  ) with respect to t.

Brennan Frye · Accepted Answer

Split the single integral into multiple integrals.

$\int - \frac{t^{3}}{2} d t + \int \frac{5}{2 t^{2}} d t + \int 2 t d t + \int - 3 \cos (t) d t$

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

$- \int \frac{t^{3}}{2} d t + \int \frac{5}{2 t^{2}} d t + \int 2 t d t + \int - 3 \cos (t) d t$

Since $\frac{1}{2}$ is constant with respect to $t$ , move $\frac{1}{2}$ out of the integral.

$- (\frac{1}{2} \int t^{3} d t) + \int \frac{5}{2 t^{2}} d t + \int 2 t d t + \int - 3 \cos (t) d t$

By the Power Rule, the integral of $t^{3}$ with respect to $t$ is $\frac{1}{4} t^{4}$ .

$- \frac{1}{2} (\frac{1}{4} t^{4} + C) + \int \frac{5}{2 t^{2}} d t + \int 2 t d t + \int - 3 \cos (t) d t$

Since $\frac{5}{2}$ is constant with respect to $t$ , move $\frac{5}{2}$ out of the integral.

$- \frac{1}{2} (\frac{1}{4} t^{4} + C) + \frac{5}{2} \int \frac{1}{t^{2}} d t + \int 2 t d t + \int - 3 \cos (t) d t$

Apply basic rules of exponents.

$- \frac{1}{2} (\frac{1}{4} t^{4} + C) + \frac{5}{2} \int t^{- 2} d t + \int 2 t d t + \int - 3 \cos (t) d t$

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

$- \frac{1}{2} (\frac{1}{4} t^{4} + C) + \frac{5}{2} (- t^{- 1} + C) + \int 2 t d t + \int - 3 \cos (t) d t$

Since $2$ is constant with respect to $t$ , move $2$ out of the integral.

$- \frac{1}{2} (\frac{1}{4} t^{4} + C) + \frac{5}{2} (- t^{- 1} + C) + 2 \int t d t + \int - 3 \cos (t) d t$

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

$- \frac{1}{2} (\frac{1}{4} t^{4} + C) + \frac{5}{2} (- t^{- 1} + C) + 2 (\frac{1}{2} t^{2} + C) + \int - 3 \cos (t) d t$

Since $- 3$ is constant with respect to $t$ , move $- 3$ out of the integral.

$- \frac{1}{2} (\frac{1}{4} t^{4} + C) + \frac{5}{2} (- t^{- 1} + C) + 2 (\frac{1}{2} t^{2} + C) - 3 \int \cos (t) d t$

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

$- \frac{1}{2} (\frac{1}{4} t^{4} + C) + \frac{5}{2} (- t^{- 1} + C) + 2 (\frac{1}{2} t^{2} + C) - 3 (\sin (t) + C)$

Simplify.

$- \frac{t^{4}}{8} - \frac{5}{2 t} + t^{2} - 3 \sin (t) + C$

Reorder terms.

$- \frac{1}{8} t^{4} - \frac{5}{2 t} + t^{2} - 3 \sin (t) + C$

Integrate − t 3 </mrow> 2  </mr

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