Trigonometric Substitution Calculator

The Trigonometric Substitution Calculator is an online tool that simplifies the evaluation of integrals involving trigonometric substitutions.

Definite Integral
Indefinite Integral

with respect to

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Lower Bound

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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.

Trigonometric Substitution Calculator with Steps

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 √a²-x², make substitution x = asin(u).
If you have √x²-a², make substitution x = asec(u).
If you have √x²+a², 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 cos²θ=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.

Related References

What is Trigonometric Substitution?
How to apply Trigonomtric Substitution Method?
How to Evaluate trigonometric substitution?
Explain trigonometric substitution:

Frequently Ask Questions

When to use trigonometric substitution in integration

Trigonometric substitution is used in integration when you have integrals that contain expressions involving the square roots of quadratic polynomials. This technique is particularly useful for simplifying integrals where the integrand includes forms.

When to Use Trigonometric Substitution:

When the integrand contains square roots of quadratic expressions. These typically take one of the following forms:

$$ \sqrt{a^2 - x^2} $$

$$ \sqrt{a^2 + x^2} $$

$$ \sqrt{x^2 - a^2} $$

When the integrand is a rational function where the denominator or the numerator contains a square root of a quadratic expression. Trigonometric Forms and Substitutions are,

For √a² - x²:

Substitution: $$ x \;=\; a\; sin (θ) $$

For √a² + x²:

Substitution: $$ x \;=\; a\; tan (θ) $$

For √x² - a²:

Substitution: $$ x \;=\; a\; sec (θ) $$

Why is trigonometric substitution so hard

Trigonometric substitution can be challenging and hard during calculation process it has several reasons that are given as:

when to use trigonometric substitution and identifying the correct form of the integral can be tricky for beginners.
The trigonometric substitution involves changing both the variable x and the differential dx, which can introduce additional steps and potential for errors.
While simplifying the integrand after substitution often requires the use of trigonometric identities and applying these identities correctly can be difficult.
After integrating, you need to convert the solution back to the original variable x. This step requires deep knowledge of inverse trigonometric functions, which can be cumbersome and error-prone.
Even after substitution, the resulting integral may still be challenging to solve and may involve complex trigonometric integrals.
Trigonometric substitution involves multiple steps: identifying the form, making the substitution, simplifying, integrating, and back substituting.Each step is different so the chance of error has been increase.

By practicing and understanding the underlying principles, you can become more comfortable and proficient with trigonometric substitution in integration.

What are the rules for finding the substitution for trigonometric functions in integrals?

When you are dealing with integrals that contain radical expressions then you need an appropriate method like trigonometric substitution.

First you need to identify the form of the integrand and then choose the appropriate substitution. The goal is to simplify the integral using trigonometric identities. Here are the rules for finding the right substitution based on the form of the integrand:

$$ \sqrt{a^2 - x^2}\; form $$

Substitution: $$ x \;=\; a\; sin (θ) $$

Trigononmetric identity is 1 − sin⁡² (θ) = cos⁡² (θ) which simplifies the square root.

$$ \sqrt{x^2 - a^2}\; form $$

Substitution: $$ x \;=\; a\; sec (θ) $$

Differential: $$ dx \;=\; a\; cos ⁡(θ)dθ\; dx $$

Reason: This substitution leverages the identity sec⁡² (θ) − 1 = tan⁡² (θ), which simplifies the square root.

$$ \sqrt{a^2 + x^2}\; form $$

Substitution: $$ x \;=\; a\; tan (θ) $$

Differential: $$ dx \;=\; a\; sec⁡^2 (θ)dθ\; dx $$

Reason: This substitution uses the identity 1 + tan² (θ) = sec⁡² (θ), which simplifies the square root.

$$ Differential: \;dx \;=\; a\; sec⁡^2 (θ)dθ\; dx $$

How is ∫1/√x^2−a^2dx calculated?

To integrate the given function,we use a trigonometric substitution method. Here's the step-by-step process:

Identify the given function that relates to which form of trigonometric substitution,

$$ \frac{1}{\sqrt{x^2 - a^2}} $$

$$ Let\; x \;=\; a\; sec (θ) $$

$$ Differentiate\; x \;=\; a\; sec (θ) $$

$$ dx \;=\; a\; sec (θ)\; tan (θ)\; dθ $$

Simplify it after putting the above substitution,

$$ \sqrt{x^2 - a^2} \;=\; \sqrt{a^2 sec^2 (θ)} - a^2 \;=\; \sqrt{a^2 (sec^2 (θ) - 1)} \;=\; a\; tan (θ) $$

$$ \int \frac{1}{\sqrt{x^2 - a^2}} dx \;=\; \int \frac{1}{a tan(θ)} . a\; sec (θ) tan (θ)\; d(θ) $$

$$ =\; \int sec (θ) dθ $$

Integrate the above function,

$$ \int sec(θ)\; dθ \;=\; ln | sec(θ) + tan(θ) | + C $$

Again replace the above expression into variable x term,

Since x = a sec(θ), use $$ sec(θ) \;=\; \frac{x}{a}\; and\; tan(θ) \;=\; \sqrt{sec^2(θ) - 1} \;=\; \sqrt{(\frac{x}{a})^2} - 1 \;=\; \frac{\sqrt{x^2 - a^2}}{a} $$

$$ ln \biggr|sec(θ) + tan(θ) \biggr| \;=\; ln \biggr| \frac{x}{a} + \frac{\sqrt{x^2 - a^2}}{a} \biggr| \;=\; ln \biggr| x + \frac{\sqrt{x^2-a^2}}{a} \biggr| $$

Therefore the solution become,

$$ \int \frac{1}{\sqrt{x^2 - a^2}} dx \;=\; ln \biggr| x + \frac{\sqrt{x^2 - a^2}}{a} \biggr| + C $$

Amanda Jepson

Last Updated February 21, 2024