This advanced integral calculator instantly simplifies definite and indefinite integrals with multiple variables. Get steps involved in the integral calculation of complicated functions with a single tap.

## What is Integral?

In calculus:

**“Integral is correlated to the sum that is used to calculate the area and volume with all generalizations”. **

Integral is the area under the graph of a function or an interval. Actually, the process of finding the integral is known as integration and it is the inverse of the derivatives that’s why it is also referred as the anti derivatives.

## How To Find Antiderivatives?

The antiderivative calculator with steps finds anti derivative of any expression with variables and also helps to realize the upper and lower bound with the maximum and minimum values of the intervals.

Our online integration calculator with steps is the best way to simplify any kind of integral. But if your goal comes up with manual calculations, you ought to grip over both definite and indefinite integration techniques.

Let us resolve a couple of examples to clarify your concept!

### Definite Integral:

Solve the following definite integral with steps

$$ \int_{0}^{1}\left( 3 x^{2} + x - 1 \right), dx $$

#### Solution:

First of all, we need to get the results for indefinite integration of the given integral

$$ \int{\left(3 x^{2} + x - 1\right),dx} $$

$$ = x \left(x^{2} + \frac{x}{2} - 1\right) $$

The fundamental theorem of definite integration states that

$$ \int{a}^{b} F\left(x\right) dx = f\left(b\right)-f\left(a\right) $$

$$ \left(x \left(x^{2} + \frac{x}{2} - 1\right)\right)|_{\left(x = 1\right)} = \frac{1}{2} $$

$$ \left(x \left(x^{2} + \frac{x}{2} - 1\right)\right)|_{\left(x = 0\right)} = 0 $$

$$ \int_{0}^{1}\left(3 x^{2} + x - 1\right), dx $$

$$ = \left(x \left(x^{2} + \frac{x}{2} - 1\right)\right)|_{\left(x=1\right)}-\left(x \left(x^{2} + \frac{x}{2} - 1\right)\right)|_{\left(x=0\right)} $$

$$ =\frac{1}{2} $$

Which is the required answer. You can also verify the results by using the smart online integral calculator in the blink of a moment.

### Indefinite Integral:

Evaluate the integral given as under

$$ \int x*cos\left(x^{2}\right), dx $$

#### Solution:

Let us make a supposition that

$$ u = x^{2} $$

Calculating the antiderivative formula of the above equation by applying the power rule:

$$ \frac{d}{dx}\left(x^{n}\right) = nx^(n-1) $$

Substitute n=2

$$ \frac{d}{dx}\left(x^{2}\right) = 2x^(2-1) $$

$$ \frac{d}{dx}\left(x^{2}\right) = 2x $$

As $$ x^{2} = u $$

so we have

$$ d\left(u\right) = \left(x^{2}\right) = 2xdx $$

$$ d\left(u\right) = xdx $$

Now, applying antiderivative rule:

$$ \int x*cos\left(x^{2}\right), dx $$

$$ = \int d\frac{cos\left(u\right)}{2}, du $$

We need to apply multiply rule which is as follows

$$ \int cf\left(u\right), du $$

$$ = c\int f\left(u\right), du $$

$$ \int \frac{cos\left(u\right)}{2}, du $$

$$ = \left(\frac{\int cos\left(u\right), du}{2}\right) $$

As the integral of cosine is given as follows

$$ \int cos \left(u\right), du = sin \left(u\right) $$

$$ int cos \left(u\right), du = \frac{sin\left(u\right)}{2} $$

As in the start, we let the

$$ u = x^{2} $$

$$ \frac{sin\left(u\right)}{2} = \frac{sin\left(x^{2}\right)}{2} $$

$$ \int x*cos\left(x^{2}\right), dx $$

$$ = \dfrac{sin\left(x^{2}\right)}{2} $$

Adding the constant of integration here which is C

$$ \int x*cos\left(x^{2}\right), dx $$

$$ = d\frac{sin\left(x^{2}\right)}{2} + C $$

Which is the required integral calculations of the given function and can also be verified by using the indefinite integral solver.

## Working of Integral Calculator:

To use our antiderivative calculator you can get the integral of any function. Just enter the following inputs and get instant integral calculations!

**Inputs:**

- Enter the function in its respective field
- Select the related variable from a neighboring list
- Select the type of integral
- If you choose “Definite Integral”, enter the lower and upper bounds
- Tap "Calculate"

**Outputs:**

Our online definite integral calculator will gives you the following answer.

- Definite and indefinite integrals
- Plots of integrals with their real and imaginary parts
- Integral simplification with steps

## FAQs:

### Can You Take Numbers Out of an Integral?

Yes definitely! You can drag the constant numbers out of the integrals to make the calculations easy.

For example, the integral $$ \int 3y + 9 $$ is as same as we multiply the number 3 by the integral \(y + 3\).

### What Is The Use Of Anti Derivatives?

This term is used to estimate the area under the curve, volume of a solid, distance, velocity, acceleration, average value of a function, and the area of any shape. For this purpose, you take help of our anti derivative calculator.

### Can an Integral be Infinite?

Yes! Any indefinite integral that is defined with positive and negative limits is said to be infinite. You can also evaluate such kind of integration with this indefinite integral calculator with steps.

### Can You Take the Integral of Every Function?

An integral can be taken of only a continuous function. The reason is that such a function is defined and displays the area under the curve.

### Can an Integral be Zero?

Yes, it is only a definite integral that can be either positive, negative, or zero.

### What Is Antiderivative of E to X?

The antiderivative of e^x is written in the form of ex + c where c is the integration constant.

### Is an Integral Always Differentiable?

You can only differentiate the integral of a continuous function which is indefinite in its nature.

### Why Do Integrals Have a Constant C?

The constant C is added to represent those functions whose derivatives are the original functions.

## Important Integral Formulas:

Functions | Integration |
---|---|

∫1 dx | x + c |

∫x^{n} dx |
x^{n+1}/ n+1 + c |

∫a dx | ax + c |

∫ (1/x) dx | lnx + c |

∫ a^{x} dx |
a^{x} / lna + c |

∫ e^{x} dx |
e^{x} + c |

∫ sinx dx | -cosx + c |

∫ cosx dx | sinx + c |

∫ tanx dx | - ln|cos x| + c |

∫ cosec^{2}x dx |
-cot x + c |

∫ sec^{2}x dx |
tan x + c |

∫ cotx dx | ln|sinx| + c |

∫ (secx)(tanx) dx | secx + c |

∫ (cosecx)(cotx) dx | -cosecx + c |

∫ 1/(1-x^{2})^{1/2} dx |
sin^{-1}x + c |

∫ 1/(1+x^{2})^{1/2} dx |
cos^{-1}x + c |

∫ 1/(1+x^{2}) dx |
tan^{-1}x + c |

∫ 1/|x|(x^{2} - 1)^{1/2} dx |
cos^{-1}x + c |

## References:

From the source **Wikipedia:** Pre-calculus integration, Terminology, Interpretations, Riemann integral, Lebesgue integral, Properties, how to find the antiderivative of a fractions?

From the source **Khan Academy: **Approximation with Riemann sums, Summation notation, Interpreting the behavior of accumulation functions, Fundamental theorem of calculus and definite integrals, Reverse power rule, find antiderivative.