To evaluate the integrals on this antiderivative calculator, follow below steps:
In a couple of seconds this integration solver gives the answer with a whole labelled solving process.
The online definite and indefinite integral calculator is a tool with an advanced algorithm that can solve complex problems like integration.
This integral calculator/solver is best because it provides solutions with steps and all possible methods for the integration. Its usability is great as it has a keyboard option to input various math operations.
This tool is also known as the definite and indefinite integral solver because it allows the user to evaluate functions using both integral methods.
Integration or antidifferentiation is the inverse of differentiation i.e the process of finding the slope. Integration is the process in which we find integrals.
Integrals are of two types; indefinite and definite. The indefinite integral gives the original function of a differentiated equation. Meaning that it undoes the differentiation.
In definite integrals, you have to find the volume during a particular time period. This is the reason you have to use upper and lower bound limits.
Since there are two types of integrations, consequently there are two methods. First, we will learn how to perform indefinite integration.
A reminder that you can always use the antiderivative calculator above to save yourself the trouble. For your help, we will discuss how to do it on paper.
Integration notation “ ∫ ”(fancy s) is applied to the function. Then the function is examined and the rules of integration are applied carefully and step by step. Famous integration rules are:
When no more evaluation is possible, an integral constant is added.
Note: The integral constant is “+c” and it is added after every integral evaluation. It is because integral accounts for all the possible derivatives of the function.
Example:
Evaluate Sin(x) + x/2.
Solution:
Step 1: Rearrange the function.
= x/2 + Sin(x)
Step 2: Apply the integral notation.
=\(\int\frac{x}{2}+\:Sin\:\left(x\right).dx\)
Step 3: Apply the sum rule.
=\(\int \frac{x}{2}.dx\:+\:\int Sin\:\left(x\right).dx\)
Step 4: Solve the first part of the function.
=\(\int \frac{x}{2}.dx+\:\int Sin\:\left(x\right).dx\)
The 2 in the denominator will remain intact according to the constant rule. But the power rule is to be applied on the numerator x. So,
=\(\frac{\int \:x.\:dx}{2}+\:\int \:Sin\:\left(x\right).dxx\)
=\(\:\frac{\int \:\frac{x^{1+1}}{1+1}\:.dx}{2}+\:\int Sin\:\left(x\right).dx\:.dx\)
=\(\frac{x^2}{2\cdot 2}+\:\int Sin\:\left(x\right).dx\:\)
=\(\frac{x^2}{4}+\:\int Sin\:\left(x\right).dx\)
Step 5: Now, apply the suitable integral rule on the trigonometric value.
By the integration rules the integral of sin(x) is -cos(x). hence,
=\(\frac{x^2}{4}-cos\:\left(x\right)\)
Step 6: Add the constant.
=\(\frac{x^2}{4}-cos\:\left(x\right)+\:c\)
This was an example of the indefinite integral. Let’s see an example of definite integral as well.
Example 2:
Solve the function above using the definite integration when the upper and lower bound limits are 2 and 1 respectively.
Solution:
Step 1: Apply the notation and solve as before.
=x/2 + Sin(x)
=\(\int _1^2\:\frac{x}{2}+\:Sin\:\left(x\right).dx\:\)
=\(\frac{x^2}{4}\int _1^2\:+\int _1^2\:Sin\:\left(x\right).dx\)
=\(\frac{x^2}{4}\int _1^2\:-cos\:\left(x\right)\int _1^2\)
Step 2: Put the upper value first and then the lower value. After that, solve.
=\(\left[\frac{\left(2\right)^2}{4}-\frac{\left(1\right)^2}{4}\right]-\left[cos\:\left(2\right)-cos\left(1\right)\right]\)
=\(\left[\frac{4}{4}-\frac{1}{4}\right]+\left[cos\left(1\right)-cos\:\left(2\right)\right]\)
=\(\frac{3}{4}+cos\left(1\right)-cos\:\left(2\right)\)
This is the definite integration of the same function. Integration by parts is quite difficult so it is suggested to use the calculator for that purpose.
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