find the Fourier transform of:  f(t) = cos(3*pi*t)[u(t+4)

a) ${\displaystyle f\left(t\right)=\cos (3\cdot t\cdot \pi )(u(t+4)-u(t-4))}$

Integrate by parts using the formula ${\displaystyle \int udv=uv-\int vdu}$, where ${\displaystyle u=u(t+4)-u(t-4)}$ and ${\displaystyle dv=\cos (3\cdot t\cdot \pi )}$.

${\displaystyle (u(t+4)-u(t-4))\left(\frac{1}{3\pi }\sin (3\cdot t\cdot \pi )\right)-\int \frac{1}{3\pi }\sin (3\cdot t\cdot \pi )\cdot 0dt}$

Simplify.

Combine ${\displaystyle \frac{1}{3\pi }}$ and ${\displaystyle \sin \left(3t\pi \right)}$.

${\displaystyle (u(t+4)-u(t-4))\frac{\sin \left(3t\pi \right)}{3\pi }-\int \frac{1}{3\pi }\sin (3\cdot t\cdot \pi )\cdot 0dt}$

Combine ${\displaystyle \frac{1}{3\pi }}$ and ${\displaystyle \sin \left(3t\pi \right)}$.

${\displaystyle (u(t+4)-u(t-4))\frac{\sin \left(3t\pi \right)}{3\pi }-\int \frac{\sin \left(3t\pi \right)}{3\pi }\cdot 0dt}$

Multiply ${\displaystyle \frac{\sin \left(3t\pi \right)}{3\pi }}$ by ${\displaystyle 0}$.

${\displaystyle (u(t+4)-u(t-4))\frac{\sin \left(3t\pi \right)}{3\pi }-\int 0dt}$

The integral of ${\displaystyle 0}$ with respect to ${\displaystyle t}$ is ${\displaystyle 0}$.

${\displaystyle (u(t+4)-u(t-4))\frac{\sin \left(3t\pi \right)}{3\pi }-(0+C)}$

Simplify the answer.

Add ${\displaystyle 0}$ and ${\displaystyle C}$.

${\displaystyle (u(t+4)-u(t-4))\frac{\sin \left(3t\pi \right)}{3\pi }-C}$

Rewrite ${\displaystyle (u(t+4)-u(t-4))\frac{\sin \left(3t\pi \right)}{3\pi }-C}$ as (ut+4u−ut+4u)sin(3tπ)3π−C+C${\displaystyle (ut+4u-ut+4u)\frac{\sin \left(3t\pi \right)}{3\pi }-C+C}$.

${\displaystyle (ut+4u-ut+4u)\frac{\sin \left(3t\pi \right)}{3\pi }-C+C}$

Simplify.

Subtract ${\displaystyle ut}$ from ${\displaystyle ut}$.

${\displaystyle (4u+0+4u)\frac{\sin \left(3t\pi \right)}{3\pi }-C+C}$

Add ${\displaystyle 4u}$ and ${\displaystyle 0}$.

${\displaystyle (4u+4u)\frac{\sin \left(3t\pi \right)}{3\pi }-C+C}$

Add ${\displaystyle 4u}$ and ${\displaystyle 4u}$.

${\displaystyle 8u\frac{\sin \left(3t\pi \right)}{3\pi }-C+C}$

Combine ${\displaystyle 8}$ and ${\displaystyle \frac{\sin \left(3t\pi \right)}{3\pi }}$.

${\displaystyle u\frac{8\sin \left(3t\pi \right)}{3\pi }-C+C}$

Combine ${\displaystyle u}$ and ${\displaystyle \frac{8\sin \left(3t\pi \right)}{3\pi }}$.

${\displaystyle \frac{u\left(8\sin \left(3t\pi \right)\right)}{3\pi }-C+C}$

Move ${\displaystyle 8}$ to the left of ${\displaystyle u}$.

$\frac{8u\sin \left(3t\pi \right)}{3\pi }-C+C$

b) ${\displaystyle f\left(t\right)=\cos (3\cdot t\cdot \pi )(u(t+4)-u(t-4))}$

Simplify each term.

Apply the distributive property.

${\displaystyle f\left(t\right)=\cos \left(3t\pi \right)(ut+u\cdot 4-u(t-4))}$

Move 4${\displaystyle 4}$ to the left of u${\displaystyle u}$.

${\displaystyle f\left(t\right)=\cos \left(3t\pi \right)(ut+4\cdot u-u(t-4))}$

Apply the distributive property.

${\displaystyle f\left(t\right)=\cos \left(3t\pi \right)(ut+4u-ut-u\cdot -4)}$

Multiply ${\displaystyle -4}$ by ${\displaystyle -1}$.

${\displaystyle f\left(t\right)=\cos \left(3t\pi \right)(ut+4u-ut+4u)}$

Simplify by adding terms.

Combine the opposite terms in ${\displaystyle ut+4u-ut+4u}$.

Subtract ${\displaystyle ut}$ from ${\displaystyle ut}$.

${\displaystyle f\left(t\right)=\cos \left(3t\pi \right)(4u+0+4u)}$

Add ${\displaystyle 4u}$ and ${\displaystyle 0}$.

${\displaystyle f\left(t\right)=\cos \left(3t\pi \right)(4u+4u)}$

Add ${\displaystyle 4u}$ and ${\displaystyle 4u}$.

${\displaystyle f\left(t\right)=\cos \left(3t\pi \right)\left(8u\right)}$

Simplify the expression.

Rewrite using the commutative property of multiplication.

${\displaystyle f\left(t\right)=8\cos \left(3t\pi \right)u}$

Reorder factors in ${\displaystyle 8\cos \left(3t\pi \right)u}$.

$f\left(t\right)=8u\cos \left(3t\pi \right)$