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 β« [sin (5π‘) π’ + cos (3π‘) π£] dπ‘.

02:09

### Video Transcript

Calculate the integral the integral of the sin of five π‘ π’ plus the cos of three π‘ π£ with respect to π‘.

The question wants us to calculate the indefinite integral of a vector-valued function. We input a value of π‘ and it outputs a position vector. We do this by integrating each of our component functions with respect to π‘. Thatβs the integral of the sin of five π‘ and the integral of the cos of three π‘. So we want to calculate the integral of the sin of five π‘ with respect to π‘ and the integral of the cos of three π‘ with respect to π‘.

We recall for a constant π, theWE integral of the sin of ππ‘ with respect to π‘ is equal to negative the cos of ππ‘ divided by π plus the constant of integration π. Using this, we can evaluate the integral of the sin of five π‘ with respect to π‘. Itβs equal to negative the cos of five π‘ divided by five plus the constant of integration we will call π one. We also know for any constant π, the integral of the cos of ππ‘ with respect to π‘ is equal to the sin of ππ‘ divided by π plus the constant of integration π. Using this, we have the integral of the cos of three π‘ with respect to π‘ is equal to the sin of three π‘ over three plus the constant of integration we will call π two.

Since weβve now found the integral of each of our component functions, we can write the integral of the sin of five π‘ π’ plus the cos of three π‘ π£ with respect to π‘ as negative the cos of five π‘ over five plus π one π’ plus sin of three π‘ over three plus π two π£. 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 π‘ over five π’ plus the sin of three π‘ over three π£ plus π one π’ plus π two π£. However, π one and π two are just constants of integration. So we could combine this entire expression into a vector we will call π.

Therefore, weβve shown the integral of our vector-valued function with respect to π‘ is equal to negative one-fifth the cos of five π‘ π’ plus the sin of three π‘ over three π£ plus π.

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