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## How to integrate y with respect to x in Calculus

Learning how to integrate y with respect to x in calculus is important for solving many different types of problems. Integrating y with respect to x means **finding the area under the curve** when y is plotted against x, so it can be useful in solving geometrical problems. Integration is also useful when solving problems in physics.

Integrating is the opposite of differentiation, so to solve the problem, we need to find a function that gives y when it is differentiated with respect to x. The rules of integration are opposite to the rules of differentiation. For example, to differentiate a power function, you need to multiply by the original exponent and then reduce the exponent by one, i.e., d/dx (x^{n}) = n x^{(n-1)}. To integrate a power function, you do the opposite, i.e., increase the exponent by one and then divide by the new exponent, so that [int] x^{n} = x^{(n+1)} / (n+1) + c.

## How to integrate y with respect to x in Calculus: Steps for power functions

**Sample problem**: Integrate y = x^{2} + 1 with respect to x, for x between 0 and 1.

Step 1: Increase the exponent of each term by one, and divide each term by the new exponent.

[int] y dx = [int] (x^2 + 1) dx = x^3 / 3 + x.

Step 2: Substitute the limits of the integration range for x.

At x = 1, x^{3} / 3 + x = 4/3

At x = 0, x^{3} / 3 + x = 0

Step 3: Find the difference between the values (i.e. subtract the values in the previous step).

The value of y integrated with respect to x for x between 0 and 1 is 4/3.

**Tip:** If the example had not given an integration range, the answer would be x^3 + x + c.

If you are calculating the indefinite integral of y with respect to x (i.e., if the range of integration is not defined), you need to add an arbitrary constant “+ c” to the end of the equation. This represents the fact that there are a range of possible functions that will differentiate with respect to x to give y, because constant terms disappear under differentiation. Alternatively, a range of integration might be given, in which case, put the given values of x into the integrated expression and find the difference between the two values, as in the example below.

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