Question Video: Calculating an Indefinite Integral of a Vector-Valued Function | Nagwa Question Video: Calculating an Indefinite Integral of a Vector-Valued Function | Nagwa

Question Video: Calculating an Indefinite Integral of a Vector-Valued Function Mathematics • Higher Education

Calculate the integral ∫ [sin (5𝑡) 𝐢 + cos (3𝑡) 𝐣] d𝑡.

02:09

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.

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