Introduction to Trigonometric Substitution Calculator
The trig substitution calculator with steps is a free tool that helps to find the solution of integral problems that cannot be solved by any other method of integration.
The trig sub calculator determines the antiderivative functions that have square roots or rational powers of quadratic expressions in a fraction of a second. Trigonometric substitution is a technique commonly used in calculus to simplify such integrals by substituting trigonometric functions for certain variables.
Our calculator can help you solve integrals more efficiently and accurately. It serves as a valuable tool for students and professionals working in calculus and related fields where such integrals are encountered frequently. Additionally, if you would like to find the solution of complex integral questions, you can utilize our U substitution calculator. Our calculator also determines the integral function who do not solve by any other method of integration in the run of time.
What is Trigonometric Substitution?
Trigonometric substitution is a process in which trigonometric functions are substitution into an original problem that has a square root or radical expression for the solution of integral questions easily.
Trigonometric substitution is a technique commonly used in calculus to simplify integrals that involve expressions with square roots or the sum/difference of squares. It is particularly useful for integrating functions where algebraic manipulation alone is insufficient.
Our calculator is a powerful technique in calculus, particularly for handling integrals involving square roots or expressions that resemble trigonometric identities. Further, if you eould like to find the antiderivative for the given function, you can utilize our integration by parts calculator. Our calculator also helps you to easily determine the special type of integration which is a product of two functions.
Formula Used by Trig Substitution Calculator
There are three expressions that take trigonometric functions to solve these types of integral problems. According to the expression, different trigonometric substitutions are used. The expression used by our integration by trigonometric substitution calculator is as follows,
$$ \sqrt{a^2 - x^2} \; x \;=\; asin \theta \; dx \;=\; acos \theta d \theta $$
$$ \sqrt{a^2 + x^2} or \sqrt{a^2 + x^2} \; x \;=\; atan \theta \; dx \;=\; asec^2 \theta d \theta $$
$$ \sqrt{x^2 - a^2} \; x\;=\; asec \theta \;dx\;=\; asec \theta tan \theta d \theta $$
Working Method of Trig Sub Calculator
The trig sub integral calculator takes specific types of antiderivative expressions and substitutes trigonometric functions in these integral problems to give instant solutions.
When you give your integral question as an input in this tool.
First, our calculator identifies the problem that whether the given integral has one of the same expressions mentioned above. After recognizing the nature of the function, change x and dx into cross-pounding trigonometric substitution.
Then simplify the function according to the rules of integration after substitution, two case finds which are:
- In case of definite integral apply upper and lower limit and get result. For further clarification, you can utilize the bounded integral calculator to solve definite integrals with specified limits, providing a comprehensive approach to integral calculus problems.
- In the case of an indefinite integral function replace trigonometric substitution from the original function which is in terms of x.
Let's take an example to see the trigonometric substitution process for an integral problem to know how this online trig substitution calculator with steps works
Example of Trigonometric Substitution:
An example of the trigonometric substitution problem is given to know how to solve such problems manually. These problems can be solved by using the trigonometric substitution calculator but it is also important to know each step so, here’s an example,
Example:
Find the following:
$$ \int_{-3}^{3} \sqrt{9 - x^2} dx $$
Solution:
$$ 9 sin^2 \theta + 9cos^2 \theta \;=\; 9 $$
Hence,
$$ 9cos^2 \theta \;=\; 9 - 9 sin^2 \theta $$
If we let,
$$ x \;=\; 3sin \theta $$
Then,
$$ 9 - x^2 \;=\; 9 - 9 sin^2 \theta \;=\; 9cos^2 \theta $$
$$ \int_{-3}^{3} \sqrt{9 - x^2} dx \;=\; \int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \sqrt{9 - 9sin^2 \theta} (3cos \theta) d \theta $$
$$ \int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \sqrt[3]{9 cos^2 \theta} cos \theta d \theta $$
$$ \int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} 3|3 cos \theta| cos \theta d \theta $$
$$ \int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} 9cos^2 \theta d \theta $$
$$ \int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \frac{9}{2} (1+cos(2 \theta)) d \theta $$
$$ -\frac{9}{2} \biggr( \theta + \frac{1}{2} sin(2 \theta) \biggr) \biggr|_{\frac{-\pi}{2}}^{\frac{\pi}{2}} -\frac{9}{2} \pi $$
Thus it is the final solution of our function with specific limits. If you would like to solve this problem further, you can use our integral indefinite calculator. Additionally, our calculator also provides a comprehensive tool for further analysis and exploration of functions without specific limits.
How to Use Trigonometric Substitution Calculator with Steps
The trig substitution calculator with steps has a simple design that you use for solving integral problems easily if you follow the given guidelines. These are:
- Enter your integral function(definite or indefinite) in the trig sub calculator.
- Select a variable from the given list(x,y,z) that you want to evaluate for integral questions.
- If the problem is related to a definite integral then add the upper and lower limits in our calculator.
- If a given question is related to an indefinite then you do not need to add limit values for the solution
- Press the “Calculate” button to get the solution of the integral problem
- Recalculate button brings you back to the new page for the new calculation
Result Obtained from Trig Sub Integral Calculator
You will obtain the solution of the integral question after you give input in the integration by trigonometric substitution calculator. It may contain as:
Our calculator serves as an educational tool for learning and practicing trigonometric substitution techniques in calculus. It helps reinforce understanding of concepts and provides a platform for exploration and experimentation.
Additionally, For more advanced calculations involving integrals, you can use our integral evaluator that provides accurate results with step-by-step explanations
- Result section provides the solution of the given antiderivative problem
- Possible steps section provides you with solutions in a step-wise process
- Plot section sketch a graph according to the result of the given integral problem
Benefits of Using Our Calculator
The trig substitution calculator gives you a lot of benefits while using this tool for solving integral questions. These benefits are:
Our Calculator also offers efficiency, accuracy, comprehensiveness, and user-friendliness, making it a valuable tool for anyone working with integration problems involving trigonometric functions. Additionally, if you're interested in exploring more mathematical tools and calculators for various applications, you can access our comprehensive collection of calculators by visiting the All Calculators page.
- It saves our time and effort in doing complex calculations of integrals
- It gives results in a fraction of a second.
- This trig sub calculator provides accurate results with steps.
- You can use this calculator to practice more and more integral examples.
- It will improve your learning experience about the trigonometric substitution method.
Conclusion
The trig sub integral calculator is a valuable tool for such integral problem who has a specific expression with radical and do not solve any other rule of integration.
The trigonometric substitution calculator with steps is a reliable tool as it provides a precise result of your given problem every time whenever you give input into it.
Explore other calculus techniques with our integral calculator partial fraction. Delve deeper into solving complex integrals by employing both trigonometric substitution and integration by parts methods.