Question Video: Integrating Trigonometric Functions | Nagwa Question Video: Integrating Trigonometric Functions | Nagwa

# Question Video: Integrating Trigonometric Functions Mathematics • Second Year of Secondary School

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Determine β«(βsin π₯ β 9 cos π₯) dπ₯.

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### Video Transcript

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

Before trying to evaluate this, it can be useful to recall some of the properties of integrals. Firstly, the integral of the sum of two or more functions is 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. These properties mean 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₯.

And now we recall the general results for the integrals of the sine and cosine functions. The indefinite integral of sin of ππ₯ is equal to negative one over π times cos ππ₯ plus the constant of integration π. And the integral of cos of ππ₯ evaluated with respect to π₯ is equal to one over π times sin ππ₯ plus the constant π. So in our case, the constant π is equal to one, and our integral is negative negative cos π₯ plus the constant π΄ minus nine times sin π₯ plus the constant π΅. And weβve chosen π΄ and π΅ to show that these are different constants of integration.

Now distributing the parentheses and combining the two constants π΄ and π΅ into a single constant πΆ, we have the indefinite integral of negative sin π₯ minus nine cos π₯ evaluated with respect to π₯ is equal to cos π₯ minus nine times sin π₯ plus the constant πΆ.

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