Video Transcript
Calculate the integral the integral
of the sin of five 𝑡 𝐢 plus the cos of three 𝑡 𝐣 with respect to 𝑡.
The question wants us to calculate
the indefinite integral of a vector-valued function. We input a value of 𝑡 and it
outputs a position vector. We do this by integrating each of
our component functions with respect to 𝑡. That’s the integral of the sin of
five 𝑡 and the integral of the cos of three 𝑡. So we want to calculate the
integral of the sin of five 𝑡 with respect to 𝑡 and the integral of the cos of
three 𝑡 with respect to 𝑡.
We recall for a constant 𝑎, theWE
integral of the sin of 𝑎𝑡 with respect to 𝑡 is equal to negative the cos of 𝑎𝑡
divided by 𝑎 plus the constant of integration 𝑐. Using this, we can evaluate the
integral of the sin of five 𝑡 with respect to 𝑡. It’s equal to negative the cos of
five 𝑡 divided by five plus the constant of integration we will call 𝑐 one. We also know for any constant 𝑎,
the integral of the cos of 𝑎𝑡 with respect to 𝑡 is equal to the sin of 𝑎𝑡
divided by 𝑎 plus the constant of integration 𝑐. Using this, we have the integral of
the cos of three 𝑡 with respect to 𝑡 is equal to the sin of three 𝑡 over three
plus the constant of integration we will call 𝑐 two.
Since we’ve now found the integral
of each of our component functions, we can write the integral of the sin of five 𝑡
𝐢 plus the cos of three 𝑡 𝐣 with respect to 𝑡 as negative the cos of five 𝑡
over five plus 𝑐 one 𝐢 plus sin of three 𝑡 over three plus 𝑐 two 𝐣. We could leave our answer like
this. However, removing the parentheses
and rearranging, we can see that our answer is equal to negative the cos of five 𝑡
over five 𝐢 plus the sin of three 𝑡 over three 𝐣 plus 𝑐 one 𝐢 plus 𝑐 two
𝐣. However, 𝑐 one and 𝑐 two are just
constants of integration. So we could combine this entire
expression into a vector we will call 𝐜.
Therefore, we’ve shown the integral
of our vector-valued function with respect to 𝑡 is equal to negative one-fifth the
cos of five 𝑡 𝐢 plus the sin of three 𝑡 over three 𝐣 plus 𝐜.
