promath The general solution of tanx=tany for x is given by x=nπ+y where n= 0, ±1, ±2, ±3, …. In this post, we will learn how to find the general solution of tanx=tany as well as tanθ=tanα. Solutions of tanx = tany are given by x=nπ+y, n ∈ ℤ Let us now solve the trigonometric equation … Read more
The simplification of the expression 1-sec2x is equal to -tan2x. In this post, we will establish 1-sec^2x formula. We have: Let us now prove the above formulas. 1-sec2x Formula The formula of 1-sec2x is given below: $\boxed{1-\sec^2 x=-\tan^2 x}$ Proof: To simplify the expression $1-\sec^2 x$, we will follow the below steps: Step 1: We will … Read more
Note that cosx cosy is the product of two cosine functions cosx and cosy. The formula of the product cosx cosy is given as follows: cosx cosy = $\dfrac{\cos(x+y)+\cos(x-y)}{2}$ Proof of cosx cosy Formula Let us now prove the cosx cosy formula cosx cosy = $\dfrac{\cos(x+y)+\cos(x-y)}{2}$ Proof: We know that cos(x+y) = cosx cosy – … Read more
The simplification of the expression 1-cos2x is equal to sin2x. In this post, we will learn how to find the formula of 1-cos^2x. 1-cos2x Formula The formula of 1-cos2x is given below: 1-cos2x = sin2x 1-cos2x = sin2x Proof To prove $1-\cos^2x =\sin^2 x$, we will follow the below steps: Step 1: At first, we will … Read more
Answer: The Laplace transform of t square is equal to 2/s3, that is, L{t2} = 2/s3. In this post, we will learn about the formula of L{t2} along with its proof. Laplace Transform of t2 Formula The Laplace of t square formula is given as follows L{t2} = 2/s3. Laplace Transform of t2 Proof We … Read more
The integration of xn is given as follows: ∫xndx = xn+1/(n+1) + C where C is an integration constant. In this post, we will learn how to find the integral of x to the power n. Integration of xn Formula The integration formula of xn (n-th power x) is given below: ∫xndx = xn+1/(n+1) + … Read more
The integration of root x + 1/root x is equal to 2/3 x3/2+2√x+C where C is an integral constant. In this post, we will learn how to integrate √x+1/√x. Integral of √x+1/√x Question: Find the integral $\int (\sqrt{x}+\dfrac{1}{\sqrt{x}})\ dx$ Answer: $\int (\sqrt{x}+\dfrac{1}{\sqrt{x}})\ dx$ = $\int \sqrt{x}\ dx$ + $\int \dfrac{1}{\sqrt{x}}\ dx$ by the sum rule … Read more
The modulus of x, that is, |x| is a continuous function. In this post, we will learn how to prove mod x is continuous. Let us now show that mod x is continuous at the point x=0. f(x)=|x| is continuous at 0 Let f(x)=|x|. Note that we have So we obtain that $\lim\limits_{x \to 0} … Read more
The derivative of the modulus of sinx is equal to (sinx cosx)/|sinx|. In this post, we will learn how to differentiate mod sinx, that is, how to find d/dx(|sinx|). The differentiation formula of mod sinx is given below and we will learn how to derive this formula in details. Mod sinx Derivative Formula The derivative … Read more
The derivative of x^sinx (x to the power sinx) is equal to xsinx[sinx/x + cosx logx]. Here we learn how to differentiate x^sinx. Derivative of xsinx Formula xsinx Derivative Formula: The formula of xsinx derivative is given by d/dx(xsinx) = xsinx[sinx/x + cosx logx] Let us now give a proof of this fact. Derivative of … Read more