Evaluate- -cos 2 x -sin 2 x - d x

Consider, I=∫cos 2x sin 2x #160; dx We know sin 2θ #160; =2cosθ #160; sinθ #160; hence sin 4x =2cos 2x sin 2x ⇒sin 4x #160; 2 =cos ...

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Byju's Answer

Standard XII

Mathematics

Integration of Trigonometric Functions

Evaluate: ∫...

Question

Evaluate: $\int cos 2 x \cdot sin 2 x \cdot d x$

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Solution

Consider, $I = \int cos 2 x sin 2 x d x$

We know
$sin 2 θ = 2 cos θ sin θ$

hence
$sin 4 x = 2 cos 2 x sin 2 x \Rightarrow \frac{sin 4 x}{2} = cos 2 x . sin 2 x$

$E = \int \frac{sin 4 x}{2} d x = \frac{1}{2} \int sin 4 x d x = \frac{1}{2} \times - \frac{1}{4} cos 4 x + C$

$E = - \frac{1}{8} \times cos 4 x + C$

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Evaluate: ∫cos2x⋅sin2x⋅dx

Consider, I=∫cos2xsin2xdxWe knowsin2θ=2cosθsinθhencesin4x=2cos2xsin2x⇒sin4x2=cos2x.sin2xE=∫sin4x2dx=12∫sin4xdx=12×−14cos4x+CE=−18×cos4x+C

Evaluate: $\int cos 2 x \cdot sin 2 x \cdot d x$