## Introduction to Trigonometric Substitution Calculator With Steps:

Trigonometric substitution calculator is an online tool that helps you **evaluate the integral functions** that have trigonometric substitution terms. It is used to determine the substitution of trigonometric functions that have radical expression integral functions.

Our trig sub calculator is a helpful source as it solves the complex function of integration easily so there wouldn't be any need to do manual calculation.

## What is Trigonometric Substitution?

Trigonometric substitution is a method of integration that is used to simplify the given integral that has the square root of quadratic expressions by converting them into **trigonometric integral** expressions.

You can choose the appropriate trigonometric substitution as per your given integral function using trigonometric identities, the integral becomes easier to solve. This technique is useful in calculus for handling complex integrals that are difficult to evaluate directly. This method is used for solving the definite and indefinite integrals problem.

## How to Solve Integration By Trigonometric Substitution?

To **solve the complex** structure of an integral use the trigonometric substitution method which involves several steps. Here’s a detailed guide to the process of trigonometric substitution.

Steps for Trigonometric Substitution:

**Step 1:** Identify the quadratic expression in the integrand from the given substitution form.

- If you have √a
^{2}-x^{2}, make substitution x = asin(u). - If you have √x
^{2}-a^{2}, make substitution x = asec(u). - If you have √x
^{2}+a^{2}, make substitution x = atan(u).

**Step 2:** Choose the appropriate trigonometric substitution as per the given integral function.

**Step 3:** As per the x value, find the differentiation of that chosen substitution value as dx.

**Step 4:** Substitute x and dx in the integral function and simplify it using trigonometric identities.

**Step 5:** Take the integral with respect to integrating variables.

**Step 6: **After solving the integration, find the substitution value to change the function into variable x.

**Step 7: **For indefinite integrals, add the constant of integration C. For definite integrals, solve the integration and convert function into x and apply limits to get a solution.

## Practical Example of Trigonometric Substitution:

The solved **example of trigonometric substitution** gives you a clear-cut idea about the working process of our trig substitution calculator with steps.

### Example: Evaluate the given integral

$$ \int \sqrt{9 - x^2} dx $$

**Solution:**

The given integral function cannot be solved directly so we apply the trigonometric substitution method as:

$$ \sqrt{a^2 - x^2}, x \;=\; asinθ,\; dx \;=\; acosθ\; dθ $$

As we have the above expression in the given integral problem. So put,

$$ x \;=\; 3\; sinθ, we\; have\; dx \;=\; 3\; cosθ\; dθ $$

Add x and dx values in the integral function.

$$ \int \sqrt{9 - x^2} dx \;=\; \int \sqrt{9 - (3\;sinθ)^2} . 3 cosθ\; dθ $$

Simplify it to get a solution,

$$ \int \sqrt{9 - x^2} dx \;=\; \int 9 \sqrt{1- sin^2 θ} . cosθ\; dθ $$

$$ \int \sqrt{9 - x^2} dx \;=\; \int 9 \sqrt{cos^2 θ} cos θ\; dθ $$

$$ \int \sqrt{9 - x^2} dx \;=\; \int 9 cos^2 θ dθ $$

As cos^{2}θ=1/2+cos2θ/2 in the above expression,

$$ \int 9 \biggr( \frac{1}{2} + \frac{1}{2} cos(2θ) \biggr) dθ $$

Integrate with respect to θ and simplify it,

$$ =\; \frac{9}{2}θ + \frac{9}{4}sin(2θ) + C $$

$$ =\; \frac{2}{9}θ + \frac{9}{4}(2sinθ\; cosθ) + C $$

$$ sin(2θ) \;=\; 2sinθ\; cosθ $$

Replace the θ with the original value in x,

$$ sin^{-1}(\frac{x}{3}) \;=\; θ $$

$$ sinθ \;=\; \frac{x}{3} . cosθ \;=\; \frac{\sqrt{9 - x^2}}{3} $$

Put these values in the above term to convert it into x form again for the solution,

$$ =\; \frac{9}{2}sin^{-1} (\frac{x}{3}) + \frac{9}{2} . \frac{x}{3} . \frac{\sqrt{9 - x^2}}{3} + C $$

$$ =\; \frac{9}{2}sin^{-1} (\frac{x}{3}) + \frac{x \sqrt{9 - x^2}}{2} + C $$

## How to Use Trig Sub Integral Calculator?

Trigonometric substitution calculator has a simple design that makes it easy for the evaluation of complex integral problems. Follow our guidelines which are given as:

- Enter the expression of the complex integral function in the given input field.
- Enter the value of the upper and lower limit in the input field (if your function is a definite integral function).
- Recheck the given complex integral expression before clicking the calculate button to start the evaluation process in the trig sub calculator.
- Click the “
**Calculate**” button to get the result of your given complex integral problem. - If you are trying our trig substitution calculator for the first time then you can use the load example to learn more about this concept.
- Click on the “Recalculate” button to get a new page for finding more example solutions to complex integral problems.

## Result from Integration by Trigonometric Substitution Calculator:

Trig sub integral calculator gives you the **solution** from a given complex integral function when you add the input into it. It includes:

**Result Option**:

When you click on the result option, the trig sub calculator gives you a solution to the given integral function.

**Possible Steps**:

When you click on it, this option will provide you with a solution where all the calculations of the trigonometric substitution process are mentioned.

## Advantages of Trig Substitution Calculator With Steps:

Trig sub integral calculator provides you with tons of advantages that help you to calculate complex integral problems and give you solutions without any trouble. These advantages are:

- It is a free-of-cost tool so you can
**use it for free**to find complex integral problem solutions without spending. - Trigonometric substitution calculator with steps is a manageable tool that can manage various types of integral problems having a square root expression.
- It gives you conceptual clarity for the integral process whenever you use it solve more examples.
- Trig sub calculator saves the time that you consume on the calculation of complex integral problems manually.
- It is a reliable tool that provides you with accurate solutions whenever you use it to calculate complex integral problems without any man-made mistakes in calculation.
- Trig substitution calculator allows you to use it multiple times for the evaluation of complex number problem.