Question

1. Find the value of integral
समाकलन का मान ज्ञात करें।
cosix
sinix.cosix
dx​

Answer 1

Answer:

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Step-by-step explanation:

The derivative of sin x is cos x, so sin(x) is an antiderivative of cos(x). The derivative of cos x is - sin x. To find a function whose derivative is sin(x) we multiply by -1 to get - cos(x). ∫ π sin(x) + cos(x) dx.

The derivative of cos(x) is -sin(x). However, the antiderivative of cos(x) is sin(x)+C.

(Math | Calculus | Integrals | Table Of)

sin x dx = -cos x + C Proof csc x dx = - ln|CSC x + cot x| + C Proof

COs x dx = sin x + C Proof sec x dx = ln|sec x + tan x| + C Proof

tan x dx = -ln|COs x| + C Proof cot x dx = ln|sin x| + C Proof

The integral of cos(2x) is (1/2)sin(2x) + C, where C is a constant.

Use the half angle formula, sin^2(x) = 1/2*(1 - cos(2x)) and substitute into the integral so it becomes 1/2 times the integral of (1 - cos(2x)) dx. Set u = 2x and du = 2dx to perform u substitution on the integral. Since dx = du/2, the result is 1/4 times the integral of (1 - cos(u)) du.

In trigonometry, we use pi (π) for 180 degrees to represent the angle in radians. Hence, sin π is equal to sin 180 or sin π = 0.