∫(cos7 x-cos8 x)/(1+2 cos5 x) d x= (A) (sin2 x)/(2)+(sin3 x)/(3)+c (B) -(sin2 x)/(2)-(sin3 x)/(3

step 1 In this question we have to find the integration of cost 7x minus cost 8x upon 1 plus 2 into cos

∫(cos7 x-cos8 x)/(1+2 cos5 x) d x= (A) (sin2 x)/(2)+(sin3...

$\int \frac{\cos 7 x-\cos 8 x}{1+2 \cos 5 x} d x=$
(A) $\frac{\sin 2 x}{2}+\frac{\sin 3 x}{3}+c$
(B) $-\frac{\sin 2 x}{2}-\frac{\sin 3 x}{3}+c$
(C) $\frac{\sin 2 x}{2}-\frac{\sin 3 x}{3}+c$
(D) none of these.

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Key Concepts

Trigonometric Identities

Trigonometric identities provide formulas that allow the rewriting of expressions involving sine and cosine functions. They include various relationships such as double-angle, half-angle, and sum-difference formulas, which are crucial for transforming complicated trigonometric expressions into simpler or more familiar forms before applying integration techniques.

Sum-to-Product Identities

The sum-to-product identities convert sums or differences of trigonometric functions, like cosine or sine, into products. This transformation is often useful in integration because it may lead to factors that can cancel or simplify the integrand, enabling the use of substitution or direct integration of standard forms.

Integration Techniques for Rational Trigonometric Functions

Integrating expressions that are rational functions of trigonometric functions involves techniques that combine algebraic manipulation with trigonometric identities. By rewriting the integrand using identities and applying substitution methods, one can transform a complex trigonometric integral into a form that matches standard integral formulas, easing the integration process.

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