Video: Integrating Trigonometric Functions

Determine ∫ (βˆ’sin π‘₯ βˆ’ 9 cos π‘₯) dπ‘₯.

01:31

Video Transcript

Determine the indefinite integral of negative sin of π‘₯ minus nine cos of π‘₯ evaluated with respect to π‘₯.

It can be useful to recall the properties of integrals before evaluating this. Firstly, we know that the integral of the sum of two or more functions is actually equal to the sum of the integrals of those respective functions. And we also know that we can take any constant factors outside of the integral and focus on integrating the expression in π‘₯ itself. This means we can rewrite our integral as negative the integral of sin π‘₯ evaluated with respect to π‘₯ minus nine times the integral of cos of π‘₯ evaluated with respect to π‘₯. This means we can rewrite our integral as shown. And this means we can recall the general results for the integral of the sine and cosine functions.

The indefinite integral of sin of π‘Žπ‘₯ is negative one over π‘Ž cos π‘Žπ‘₯ plus that constant 𝑐. And the indefinite integral of cos of π‘Žπ‘₯ is one over π‘Ž times sin of π‘Žπ‘₯ plus 𝑐. So our integral is negative negative cos of π‘₯ plus 𝐴 minus nine times sin of π‘₯ plus 𝐡. Then I’ve chosen 𝐴 and 𝐡 here to show that these are different constants of integration. Distributing the parentheses and combining our constants in to one constant, we find the integral of negative sin π‘₯ minus nine cos of π‘₯ evaluated with respect to π‘₯ to be cos of π‘₯ minus nine sin of π‘₯ plus 𝑐.

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