promath The integral of lnx is equal to ∫ln(x)= xln x -x+C where C is an integral constant. Here we will learn how to integrate lnx, that is, find ∫ln(x). The integral formula of lnx is given below: ∫ln(x) dx = xln(x) -x+C where ln(x) = logex. Integration of lnx To find the integration of lnx, … Read more
The Derivative of (ax+b)/(cx+d) is equal to (ad-bc)/(cx+d)2. In this post, we will learn how to differentiate the quotient function (ax+b)/(cx+d). The derivative of (ax+b)/(cx+d) is denoted by the symbol $\dfrac{d}{dx}\left( \dfrac{ax+b}{cx+d}\right)$ and it is equal to $\dfrac{d}{dx}\left( \dfrac{ax+b}{cx+d}\right)$ $=\dfrac{ad-bc}{(cx+d)^2}$ when the denominator cx+d is nonzero, that is, x ≠ -d/c. How to Differentiate (ax+b)/(cx+d) … Read more
Derivative of absolute value of x. The derivative of mod x is denoted by d/dx(|x|) and it is equal to x/|x| for all nonzero values of x. In this post, we will learn how to differentiate modulus x. Recall that mod x is defined as below. $|x|=\begin{cases} x, & \text{ if } x\geq 0 \\ … Read more
The integration of (ax+b)/(cx+d) is equal to ax/c + (bc-ad)/c2 ln|cx+d|+K where K is an integral constant and ln denotes the natural logarithm. In this post, we will learn how to integrate the function (ax+b)/(cx+d). Integration of (ax+b)/(cx+d) Formula The integration of (ax+b)/(cx+d) formula is given below. $\int \dfrac{ax+b}{cx+d} \ dx$ $=\dfrac{ax}{c} + \dfrac{bc-ad}{c^2} \ln|cx+d|+K$, … Read more
The integration of 1/x is equal to ln|x|+C where ln denotes the natural logarithm, that is, ln|x|=loge|x| and C is the integration constant. In this post, we will learn how to integrate 1/x. Integration of 1/x Formula The formula for the integration of 1/x (1 divided by x) is given below. ∫ 1/x dx = … Read more
The integration of 2x is equal to 2x/ln2 where ln denotes the natural logarithm, that is, ln=loge. In this post, we will learn to integrate 2 to the power x with respect to x. Integration of 2x Formula The integration formula of 2x (2 to the x) with respect to x is given below. ∫ … Read more
The derivative of xx is equal to xx(1+ln x). That is, d/dx(xx)=xx(1+ln x). This can be proved by the logarithmic differentiation. Here, ln=loge. In this post, we will learn how to find the derivative of x to the x. Derivative of xx Formula The formula of xx derivative is given by d/dx(xx)=xx(1+ln x) Derivative of … Read more
The integration of 1/(1+x2) is tan-1x+C. In this post, we will learn how to integrate 1 divided by 1 plus x square by the substitution method. Integration of 1/(1+x2) Formula 1/(1+x2) integral formula: The integration formula of 1/(1+x^2) is provided below. ∫ 1/(1+x2) dx = tan-1x+C. Integration of 1/(1+x2) Proof Now, we will prove that … Read more
The integration of cosec x is ln|cosec x-cot x|. Here ln stands for the natural logarithm, that is, ln x = logex. In this post, we will learn how to integrate cosec x. Integration of Cosecx Formula Cosecx integration formula: The integration formula of cosec x is given below. ∫cosecx dx = ln|cosecx-cotx|+C where C … Read more
The integration of secx is equal to ln|secx+tanx| where ln denotes the natural logarithm, that is, the logarithm with base e. In this post, we will learn how to find the integral of secx dx. Integration of Secx Formula The sec x integration formula is given below. ∫secx dx = ln|secx+tanx|+C Integration of Secx Proof … Read more