Integral sin^2x-cos^2x/sinx cosx

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Integral sin^2x-cos^2x/sinx cosx

Something went wrong. Wait a moment and try again. Last updated at Dec. 20, 2019 by Teachoo Transcript Ex 7.3, 20 Integrate the function cos2 /(cos + sin )^2 1cos2 /(cos + sin )^2 =1(cos^2 - sin^2 )/(cos + sin )^2 =1(cos - sin )(cos + sin )/(cos + sin )^2 =1(cos - sin )/(cos + sin ) Let cos +sin = Differentiating w.r.t. x ( 2 = ^2 - ^2 ) -sin +cos = / (cos -sin ) = =1/((cos - sin ) ) Thus, our equation becomes =1((cos - sin ))/ ? /(cos - sin ) =11/ =log| |+ Putting value of = + = | + |+ We make use of the trigonometry double angle formulas, to derive this identity:We know that, (sin 2x = 2 sin x cos x)--------(i)cos 2x = cos2 x - sin2 x= 2 cos2 x - 1 [because sin2x + cos2 x = 1]----(ii)= 1 - 2 Sin2x-----------------------------(iii)We want to find the value of sin 2x cos 2x. To do this, multiply equation (i) and (ii).Sin 2x = 2 sin x cos xCos 2x = 2 cos2x - 1Multiply the above two answers to get the value:sin 2x cos 2x = (2 sin x cos x) (2 cos2x - 1)= 2 cos x (2 sin x cos2 x - sin x)Now, consider equation (i) and (iii),sin 2x = 2 sin x cos xcos 2x = 1 - 2 sin2xMultiply them to get,sin 2x cos 2x = 2 sin x cos x (1 - 2 Sin2x)= 2 cos x (sin x ? 2 sin3 x)Value of Sin 2x Cos 2xSin 2x Cos 2x = 2 Cos x (2 Sin x Cos2 x - Sin x)(or) Sin 2x Cos 2x = 2 Cos x (Sin x ? 2 Sin3 x)Integral of Sin 2x Cos 2x (Sin 2x Cos 2x) = (Sin 2x)2/ 4 + CProof:Consider sin 2x = yThen dy/dx = 2 cos 2x (or) dx = dy / 2 cos 2xNow, y cos 2x dx = y ? cos(2x) ? dy / 2 cos 2xCancel out cos 2x.y Cos(2x)dx = (y ? dy/2)= ? [ y dy ]= ? y?/2 + c= y?/4 + CTherefore, the integral of sin 2x cos 2x is (Sin 2x Cos 2x) = (Sin 2x) 2 / 4 + CDerivative of Sin 2x Cos 2xd/dx (Sin 2x Cos 2x) = 2Cos(4x)Proof:Sin 2x cos 2x = ? (2 sin 2x cos 2x) (Or) ? sin 4x By differentiating the given function:d/dx [ ? sin 4x ] = ? [d/dx (sin 4x)]= ? [cos 4x d/dx(4x) ]= ? [cos (4x) (4) ]Therefore, the derivative of sin 2x cos 2x is d/dx (Sin 2x Cos 2x) = 2 Cos (4x)Solved ExamplesExample 1: Derive the derivative of sin 2x cos 2xSolution:Sin 2x cos 2x = ? (2 sin 2x cos 2x) (Or) ? sin 4x By differentiating the given function:d/dx [ ? sin 4x ] = ? [d/dx (sin 4x)]= ? [cos 4x d/dx(4x) ]= ? [cos (4x) (4) ]Therefore, the derivative of sin 2x cos 2x is d/dx (Sin 2x Cos 2x) = 2 Cos (4x)Example 2: Derive the integral of sin 2x cos 2xSolution:Consider sin 2x = yThen dy/dx = 2 cos 2x (or) dx = dy / 2 cos 2xNow, y cos 2x dx = y ? cos(2x) ? dy / 2 cos 2xCancel out cos 2x.y Cos(2x)dx = (y ? dy/2)= ? [ y dy ]= ? y?/2 + c= y?/4 + CTherefore the integral of sin 2x cos 2x is (Sin 2x Cos 2x) = (Sin 2x) 2 / 4 + C. This website uses cookies to ensure you get the best experience. By using this website, you agree to our Cookie Policy. Learn more integral of sin^2(x)cos(x) \bold{\mathrm{Basic}} \bold{\alpha\beta\gamma} \bold{\mathrm{AB\Gamma}} \bold{\sin\cos} \bold{\ge\div\rightarrow} \bold{\overline{x}\space\mathbb{C}\forall} \bold{\sum\space\int\space\product} \bold{\begin{pmatrix}\square&\square\\\square&\square\end{pmatrix}} \bold{H_{2}O} \square^{2} x^{\square} \sqrt{\square} throot[\msquare]{\square} \frac{\msquare}{\msquare} \log_{\msquare} \pi \theta \infty \int \frac{d}{dx} \ge \le \cdot \div x^{\circ} (\square) |\square| (f\:\circ\:g) f(x) \ln e^{\square} \left(\square\right)^{'} \frac{\partial}{\partial x} \int_{\msquare}^{\msquare} \lim \sum \sin \cos \tan \cot \csc \sec \alpha \beta \gamma \delta \zeta \eta \theta \iota \kappa \lambda \mu u \xi \pi \rho \sigma \tau \upsilon \phi \chi \psi \omega A B \Gamma \Delta E Z H \Theta K \Lambda M N \Xi \Pi P \Sigma T \Upsilon \Phi X \Psi \Omega \sin \cos \tan \cot \sec \csc \sinh \cosh \tanh \coth \sech \arcsin \arccos \arctan \arccot \arcsec \arccsc \arcsinh \arccosh \arctanh \arccoth \arcsech + - = \div / \cdot \times < > \le \ge (\square) [\square] \:\longdivision{} \times \twostack{}{} + \twostack{}{} - \twostack{}{} \square! x^{\circ} \rightarrow \lfloor\square\rfloor \lceil\square\rceil \overline{\square} \vec{\square} \in \forall otin \exist \mathbb{R} \mathbb{C} \mathbb{N} \mathbb{Z} \emptyset \vee \wedge eg \oplus \cap \cup \square^{c} \subset \subsete \superset \supersete \int \int\int \int\int\int \int_{\square}^{\square} \int_{\square}^{\square}\int_{\square}^{\square} \int_{\square}^{\square}\int_{\square}^{\square}\int_{\square}^{\square} \sum \prod \lim \lim _{x\to \infty } \lim _{x\to 0+} \lim _{x\to 0-} \frac{d}{dx} \frac{d^2}{dx^2} \left(\square\right)^{'} \left(\square\right)^{''} \frac{\partial}{\partial x} (2\times2) (2\times3) (3\times3) (3\times2) (4\times2) (4\times3) (4\times4) (3\times4) (2\times4) (5\times5) (1\times2) (1\times3) (1\times4) (1\times5) (1\times6) (2\times1) (3\times1) (4\times1) (5\times1) (6\times1) (7\times1) \mathrm{Radians} \mathrm{Degrees} \square! ( ) % \mathrm{clear} \arcsin \sin \sqrt{\square} 7 8 9 \div \arccos \cos \ln 4 5 6 \times \arctan \tan \log 1 2 3 - \pi e x^{\square} 0 . \bold{=} + \mathrm{partial\:fractions} \mathrm{substitution} \mathrm{long\:division} \mathrm{trigonometric\:substitution} \mathrm{by\:parts} See All area asymptotes critical points derivative domain eigenvalues eigenvectors expand extreme points factor implicit derivative inflection points intercepts inverse laplace inverse laplace partial fractions range slope simplify solve for tangent taylor vertex geometric test alternating test telescoping test pseries test root test Related ? Graph ? Number Line ? Examples ? Our online expert tutors can answer this problem Get step-by-step solutions from expert tutors as fast as 15-30 minutes. Your first 5 questions are on us! In partnership with You are being redirected to Course Hero I want to submit the same problem to Course Hero Examples integral-calculator \int sin^{2}(x)cos(x)dx en Feedback

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