(1/4)x^2(2logx-1) + c
d/dx ( log x) = 1/x
To integrate ( x \sec x ), you can use integration by parts. Let ( u = x ) and ( dv = \sec x , dx ). Then, ( du = dx ) and ( v = \ln |\sec x + \tan x| ). Applying the integration by parts formula, you get: [ \int x \sec x , dx = x \ln |\sec x + \tan x| - \int \ln |\sec x + \tan x| , dx + C ] where ( C ) is the constant of integration. The second integral may require further techniques to evaluate.
The anti-derivative of ( \ln x ) can be found using integration by parts. Let ( u = \ln x ) and ( dv = dx ), then ( du = \frac{1}{x} dx ) and ( v = x ). Applying integration by parts, we get: [ \int \ln x , dx = x \ln x - \int x \cdot \frac{1}{x} , dx = x \ln x - x + C, ] where ( C ) is the constant of integration. Thus, the anti-derivative of ( \ln x ) is ( x \ln x - x + C ).
-logx=21.1 logx=-21.1 e^-21.1=x
To integrate ( x \tan(x) ), we can use integration by parts. Let ( u = x ) and ( dv = \tan(x) , dx ). This gives ( du = dx ) and ( v = -\ln|\cos(x)| ). Applying the integration by parts formula ( \int u , dv = uv - \int v , du ), we obtain: [ \int x \tan(x) , dx = -x \ln|\cos(x)| + \int \ln|\cos(x)| , dx + C ] The integral ( \int \ln|\cos(x)| , dx ) does not have a simple closed form, so the final result may be expressed in terms of this integral along with the logarithmic term.
d/dx ( log x) = 1/x
X(logX-1) + C
To integrate ( x \sec x ), you can use integration by parts. Let ( u = x ) and ( dv = \sec x , dx ). Then, ( du = dx ) and ( v = \ln |\sec x + \tan x| ). Applying the integration by parts formula, you get: [ \int x \sec x , dx = x \ln |\sec x + \tan x| - \int \ln |\sec x + \tan x| , dx + C ] where ( C ) is the constant of integration. The second integral may require further techniques to evaluate.
logx^2=2 2logx=2 logx=1 10^1=x x=10
The integral of f'(x) = 1 is f(x) = x + c
The anti-derivative of ( \ln x ) can be found using integration by parts. Let ( u = \ln x ) and ( dv = dx ), then ( du = \frac{1}{x} dx ) and ( v = x ). Applying integration by parts, we get: [ \int \ln x , dx = x \ln x - \int x \cdot \frac{1}{x} , dx = x \ln x - x + C, ] where ( C ) is the constant of integration. Thus, the anti-derivative of ( \ln x ) is ( x \ln x - x + C ).
∫ d/dx f(x) dx = f(x) + C C is the constant of integration.
-logx=21.1 logx=-21.1 e^-21.1=x
y=logx y=10 logx= 10 10logx = 10log1 logx = log1 x = 1 //NajN
To integrate ( x \tan(x) ), we can use integration by parts. Let ( u = x ) and ( dv = \tan(x) , dx ). This gives ( du = dx ) and ( v = -\ln|\cos(x)| ). Applying the integration by parts formula ( \int u , dv = uv - \int v , du ), we obtain: [ \int x \tan(x) , dx = -x \ln|\cos(x)| + \int \ln|\cos(x)| , dx + C ] The integral ( \int \ln|\cos(x)| , dx ) does not have a simple closed form, so the final result may be expressed in terms of this integral along with the logarithmic term.
The integration formulas covered in the second PUC syllabus primarily include basic integration techniques such as integration of power functions, trigonometric functions, exponential functions, and logarithmic functions. Key formulas include ∫ x^n dx = (x^(n+1))/(n+1) + C for n ≠ -1, ∫ sin(x) dx = -cos(x) + C, and ∫ e^x dx = e^x + C. Additionally, students learn about integration by substitution and integration by parts. Understanding these fundamental formulas is essential for solving various problems in calculus.
∫ sin(x) dx = -cos(x) + CC is the constant of integration.