(i) Evaluate the integral ∫sin2x1+cosxdx Step 1: Rewrite sin2x using the identity:sin2x=1−cos2xThus, we can rewrite the integral as:∫1−cos2x1+cosxdx Step 2: Factor the numerator:Using the difference of squares, we have:1−cos2x=(1−cosx)(1+cosx)Substituting this back in...
Find the indefinite integral∫sin xcos xdxusing the given method. Explain how your answers differ for each method.(a) Substitution where u=sin x(b) Substitution where u=cos x(c) Integration by parts(d) Using the identity sin 2x=2sin xcos x 相关知识点: 试题来源: 解析 (a) 12sin^2x+...
Step 2: Substituted(tanx)into the integral Now we substituted(tanx)into the integral: ∫sin(2x)d(tanx)=∫sin(2x)sec2xdx Step 3: Use the identity forsin(2x) We can use the double angle identity for sine: sin(2x)=2sinxcosx So, we can rewrite the integral as: ...
Question: ∫sin3(x)cos5(x)dx Integration in Calculus: Integration techniques can be used to find antiderivatives of a trigonometric function. Sometimes trigonometric identities may be needed to do so. To solve this problem, we'll use the trig-identity(sin2x=1−cos2x)and then w...
Evaluate the integral shown below using the given method. Integral of sin(x) cos(x) dx. A) Substitution where u = sin(x) B) Substitution where u = cos(x) C) Integration by parts D) Using the identity Find an appropriate substitution to evaluate the integral ...
You can also use a form of the triple angle identity: cos3(2x) = (1/4)cos(6x) + (3/4)cos(3x) All these will allow you to simplify the integral into a series of simple cosine integrals. Upvote • 1 Downvote Add comment Doug C. answered • 7d Tutor 5.0 (1,512) Math...
∫sin(θ)2cos(θ)dθ Evaluate 3(sin(θ))3+С Differentiate w.r.t. θ cos(θ)(sin(θ))2
The following elementary integral should be employed to solve for the solution of the specified indefinite integral: ∫cosxdx=sinx+C The given integrand should be simplified first. Answer and Explanation:1 Plugging in1−sin2x=cos2x, which is a trigonometric identity, to si...
∫ 1/(1 + sin2x) dx = ∫ 1/(1 + 2sinxcosx) dx = ∫ 1/[cos²x(sec²x + 2tanx)] dx = ∫ 1/(tan²x + 2tanx + 1) d(tanx)= ∫ 1/(tanx + 1)² d(tanx)= - 1/(tanx + 1) + C ∫ 1/(1 + cos2x) dx = ∫ 1/(2cos²x)...
To integrate cos x and sin x over cos x plus sin x, you can use the trigonometric identity cos^2x + sin^2x = 1 to simplify the integral. This will result in the integral of 1 over cos x plus sin x, which can be solved using the substitution method mentioned ...