Integral Rules

Introduction to integral rules:

           Integration is the process of finding a definite or indefinite integral. The outcome of integration is known as the integral of a function f(x) with respect to x. The integration process is denoted as ∫ f(x) dx . Here, f(x) is called as integrand. Integration is used in finding the area enclosed by a given surface.

 

General Formulas for integral rules

 

1) ∫ a dx = ax + c.

2) ∫ a f(x) dx = a ∫ f(x) dx

3) ∫ xn = (x(n + 1)/ n + 1) + c. Here, n ≠ -1.

4) ∫ (x + y - z) dx = ∫ x dx + ∫ y dx - ∫ z dx.

5) ∫ ex dx = ex + c.

6) ∫ au du = (au/logu) + c.

7) ∫ (1/ (au + b)) du = (1/a) log (au + b) + c.

8) ∫ (1/ (a2 + b2)) dx = (1/a) tan-1 (x/a) + c.

Integration by parts:

1) ∫ udv = uv - ∫ v du.

2) ∫ F(z) dx = ∫ F(z)/ z' dz.

3) ∫ (1/u) du = ln u + c.

Indefinite integral rules:

1) ∫ [f(x) + g(x)] dx = ∫ f(x) dx + ∫ g(x) dx.

2) ∫ [f(x) - g(x)] dx = ∫ f(x) dx - ∫ f (x) dx.

3) ∫[a f(x) + b g(x)] dx = a ∫ f(x) dx + b ∫ g(x) dx. 

Integral rules for Trigonometric functions

 

1) ∫ sinx dx = - cosx + c.

2) ∫  cosx dx = sinx + c.

3) ∫ tanx dx = ln (secx) + c.

4) ∫ cotx dx = ln (sinx) + c.

5) ∫ sec2x dx = tanx + c.

6) ∫ secx tanx dx = secx + c.

7) ∫ cscx cotx dx = -cscx + c.

8) ∫ tan2x dx = tanx - x + c.

9) ∫ sin2x dx = (1/2)x - (1/4)sin2x + c.

10) ∫ cos2x dx  = (1/2)x + (1/4)sin2x + c.

Integral rules for hyperbolic functions:

1) ∫ sinhx dx = coshx + c.

2) ∫ coshx dx = sinhx + c.

3) ∫ tanhx dx = ln coshx + c.

4) ∫ cothx dx = ln sinhx + c.

5) ∫ sech2x dx = tanhx + c.

6) ∫ csch2x dx = - cothx + c.

7) ∫ sechx dx = sin-1(tanhx) + c.

8) ∫ sinh2x dx = (sinh 2x)/4 - x/2 + c.