Subject: Calculus

# Sum Rule In Integration

The sum rule in integration is a mathematical statement or "law" that governs the mechanics involved in doing differentiation in a sum. The statement mandates that given any two functions, sum of their integrals is always equal to the integrals of their sum. Suppose we have two functions f and g, then the sum rule is expressed as;

\int [f(x) + g(x)] dx = \int f(x)dx + \int g(x)dx

for indefinite integrals and;

\int_{a}^{b} [f(x) + g(x)] dx = \int_{a}^{b} f(x)dx + \int_{a}^{b} g(x)dx

for definite integrals given that both functions are well-defined at the interval [a, b]. Notice that he sum rule applies to both definite integrals and indefinite integrals as given above.

## Alternative expression

Some books has different expression with regards to this rule but basically they are just the same. Suppose we have a function h which can be expressed as h=g+f. Then the sum rule maintains that;

\int [h(x)] dx = \int f(x)dx + \int g(x)dx

for indefinite integrals and;

\int_{a}^{b} [h(x)] dx = \int_{a}^{b} f(x)dx + \int_{a}^{b} g(x)dx

for definite integrals given that both functions are well-defined at the interval [a, b].

## Another alternative expression

Even though the rule explicitly says "sum" rule, it does literally mean that it is exclusive to integrals with addition operation only. Remember that subtraction can also be expressed as an addition and is also therefore governed by the sum rule. All assumptions above imply such that the sum rule is also expressed as;

\int [f(x) - g(x)] dx = \int f(x)dx - \int g(x)dx

for indefinite integrals and;

\int_{a}^{b} [f(x) - g(x)] dx = \int_{a}^{b} f(x)dx - \int_{a}^{b} g(x)dx

for definite integrals since f(x)-g(x)=f(x)+[-g(x)].

### Remark:

Even though we express the sum as the sum of two functions g and f in our expressions above, sum rule does not limit itself to sum of two functions only. The rule extends to sums of infinite number of functions as long as they are well-behaved.

### Example #1

Find the integral of f(x)=x^2+2x

Solution: We know that

\int x^2dx=\frac{1}{3}x^3+C

and

\int 2xdx=x^2+C.

Thus, according to the sum rule,

\int [x^2+2x]dx=\frac{1}{3}x^3+x^2

\frac{1}{3}x^3+x^2+C

### Example #2

Integrate: f(x)=4x^3+2x+1

Solution: We know that

\int 4x^3dx=x^4+C

and

\int 2xdx=x^2+C

and

\int 1dx=x+C

Thus, according to the sum rule,

\int [4x^3+2x+1]dx=x^4+x^2+x+C

x^4+x^2+x+C

### Example #3

Solve for the anti-derivative of f(x)=x - 1

Solution: We know that

\int xdx=\frac{1}{2}x^2+C

and

\int 1dx=x+C.

Thus, according to the sum rule,

\int [x+1]dx=\frac{1}{2}x^2+x+C

\frac{1}{2}x^2+x+C

### Example #4

Find the derivative of f(x)=\cos{x}+k

Solution: We know that

\int \cos{x}dx=\sin{x}+C

and

\int Cdx=kx+C.

Thus, according to the sum rule,

\int [\sin{x}+k]dx=\sin{x}+kx

### Example #5

Calculate the derivative of f(x)=\ln{x} + e^x+\cos{x}

Solution: We know that

\int \ln{x}dx=x\ln{x}-x+C

and

\int e^xdx=e^x +C

and

\int \cos{x} dx=\sin{x}+C

Thus, according to the sum rule,

\int [x^2+2x]dx=x\ln{x}-x+\sin{x}+e^x+C

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