Integrals

Integration of x/(1+x) | Integral of x/(1+x)

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The integration of x/(1+x), that is, the integral of x divided by 1+x is equal to x – ln|1+x|+C. In this post, we will learn how to integrate x/(1+x). To find the integral $\int \dfrac{x}{1+x} dx$, we will use the following power rule of the integration formula: ∫dx/x =ln|x|+C …(I) Integration of x/(1+x) Question: Find … Read more

Integration of x/(1+x^2) | Integral of x/(1+x^2)

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The integration of x/(1+x^2), that is, the integral of x divided by 1+x2 is equal to 1/2 ln(1+x2) + C, where C is a constant. Here ln denotes the natural logarithm. In this post, we will learn how to integrate x divided by 1+x^2. Let’s find the integral: $\int \dfrac{x}{1+x^2} dx$. To find the above … Read more

Integration of root 1+sin2x | ∫√(1+sin2x) dx

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The integration of the square root of 1+sin2x is given by ∫√(1+sin2x) dx = cosx-sinx+C where C is an integration constant. In this post, we will learn how to integrate square root of 1+sin2x. Integral of Square Root of 1+sin2x Question: Find the integral of 1+sin2x, that is, find ∫√(1+sin2x) dx. Answer: We will use … Read more

Integral of e^(x^2) | Integration of e^(x^2)

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 The integral of e^(x^2), that is, the integration of e to the x2 is equal to $\dfrac{\sqrt{\pi}}{2}$ erfi(x)+C, where erfi(x) is called the “imaginary error function” and C is an integration constant. In this post, we will learn to integrate the function $e^{x^2}$. Integral of $e^{x^2}$ We know that ex can be written as ex … Read more

Integral of e^2x | e^2x Integration

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The integration of e2x is equal to e2x/2. In this post, we will learn how to integrate e to the 2x. The integration of e2x can be derived directly from the formula of the integral of emx which is given below: ∫emx dx = $\dfrac{e^{mx}}{m}$ +C where C is an integration constant. Putting m=2 in … Read more

Integration of x^n: Formula, Proof | Integral of x^n

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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

Integration of root(x)+1/root(x) | Anti Derivative of √x+1/√x

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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

Integral of lnx: Formula, Proof

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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

Integration of (ax+b)/(cx+d): Formula, Proof

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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

Integration of 1/x | Integral of 1/x

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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