Use the formula ∫ f⁻¹(x) dx = x f⁻¹(x) - ∫ f(y) dy, y = f⁻¹(x)...

Question

Use the formula ∫ f⁻¹(x) dx = x f⁻¹(x) - ∫ f(y) dy, y = f⁻¹(x)

To evaluate the integrals in Exercises 77-80. Express your answers in terms of x.

∫ arctan x dx

Accepted Answer

Hello there. Today we are going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Evaluate the integral of inverse cosine of 3 x plus 1 dx. Using the formula, the integral of inverse f xDX is equal to x multiplied by inverse f x. Minus the integral of fydy where y is equal to inverse f of x, express your answer in terms of x. OK. So, it appears we have an integral that we're asked to evaluate. So we're asked to evaluate the integral of inverse cosine of 3 x plus 1 dx using the formula that is provided to us by the prom itself, and we're also told what y is equal to, where Y is equal to inverse f x, and we're ultimately asked to express our final answer in terms of X. So now that we know what we're ultimately trying to solve for. Let us first note that based on our prior knowledge, in order to evaluate the integral of the inverse cosine function using the provided formula, we must first identify what the components of the function and its inverse are. So we need to identify the components of our function and its inverse function in order to evaluate our integral of this inverse cosine function. So, with that in mind, keeping that in mind, our first step that we need to take is we need to identify our functions, starting with noting what our formula is. So we're told that our formula is the integral of inverse F of XDX is going to be equal to x multiplied by inverse. F X minus the integral of FYDY. And we are told that F, so once again, we're told that this inverse function of S, so inverse F of X is equal to inverse cosine of 3 X plus 1. So, this is what we're told by the problem itself. So with that in mind, let us state that Y be equal to inverse cosine of 3 x plus 1, which will mean that by definition of inverse functions, that will mean that cosine of y will be equal to 3 x plus 1. Fantastic. So keeping that in mind, we can now therefore safely write that cosine. Of Y, so cosine of Y minus 1 will be equal to 3X. So once again, cosine of Y minus 1 will be equal to 3X. So, when we solve for X, we will find that X is equal to cosine of Y minus 1 divided by 3. So therefore, Fy will be equal to 1 divided by 3 multiplied by cosine of y minus 1/3. And let's make a note here that since Y is equal to inverse F of X, that will mean that F of Y will be equal to simply X. Fantastic. So with that said. Our second step now is we need to substitute our values back into the formula. So now, at this point, we need to plug these components into our given formula, and when we do that, we will be able to write that the integral of inverse cosine of 3 x plus 1. DX. Will be equal to x multiplied by inverse cosine of 3 x + 1 minus the integral of parentheses of 1 divided by 3 multiplied by cosine of y. -1 divided by 3. DY fantastic, where 1 divided by 3 could be set as 1/3 2. So, our third step now is we need to evaluate the integral in terms of Y. So we now need to perform the integration for the Y term. So we need to take the integral of parentheses of 1/3 multiplied by cosine of Y minus 1/3 DY, which will be equal to 1/3 multiplied by sin of y. 1/3 multiplied by y plus c or c denotes some constant variable. So with that in mind, we now need to substitute this expression back into our main equation. So you take the integral of inverse cosine of. 3 X plus 1 DX, so 3 X plus 1 DX, fantastic, and this will be equal to X multiplied by inverse cosine of 3 X plus 1 minus square brackets of. 1/3 multiplied by sin of y minus 1/3 multiplied by y plus C. Which will mean that this will be equal to x multiplied by inverse cosine of 3 x + 1 minus 1/3 multiplied by sin of y, so 1/3 multiplied by sin of y plus 1/3 multiplied by y plus C. Fantastic. So now for our 4th step, we now need to convert back to terms of X. So with that in mind, let us recall that we stated that Y is equal to. Inverse cosine of 3 x plus 1. So considering that y is equal to inverse cosine of 3 x plus 1, we need to convert sin of y back to x. And in order to do that, we need to recall and use the identity that states that sin squared of y plus cosine squared of y will be equal to 1. Which will mean that sin, once again, this will mean that sin of y will be equal to the square root of 1 minus cosine squared. Of why. And we need to note that we took the positive root, since the range of Y equals inverse cosine of 3 x + 1 is square brackets of 0, i, so let's make a note of that. So by definition, that will mean that Y is an element of the closed interval 0, pi. And that sin of y is positive on this interval. So, once again, noting that cosine of y is equal to drum roll 3 x plus 1, that means we can safely go ahead and write that. Sin of y will be equal to the square root of 1 minus 3 x + 1 to the power of 2. Fantastic. So now we need to substitute everything back into our expression, so we can go ahead and write that x multiplied by inverse cosine of 3 x + 1 minus 1/3 multiplied by the square root of 1 minus 3x + 1 to the power of 2. Plus 1/3 multiplied by inverse cosine of 3 x + 1 + c. So that will mean for our fifth and final step is when we group, so once again now we're grouping our inverse cosine of 3 x plus 1 terms, and when we group these terms, we can go ahead and write that the integral of Inverse cosine of 3 x + 1. D X Will be Equal to, so this will be equal to. Parentheses x + 1/3 multiplied by inverse cosine of 3 x + 1 minus 1/3 multiplied by the square root of 1 minus 3 x + 1 parentheses to the power of 2. Plus C and this is our correct and final answer. Hooray, we did it. So going back to look at the prom itself, we can safely say without a doubt that our final answer is once again drum roll. Parentheses of X + 1/3 multiplied by inverse cosine of 3x + 1 minus 1/3 multiplied by the square root of 1 minus parentheses of 3 x + 1 to 2 + C. And that's it. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Use the formula ∫ f⁻¹(x) dx = x f⁻¹(x) - ∫ f(y) dy, y = f⁻¹(x)To evaluate the integrals in Exercises 77-80. Express your answers in terms of x.∫ arctan x dx

Watch next

Use the formula ∫ f⁻¹(x) dx = x f⁻¹(x) - ∫ f(y) dy, y = f⁻¹(x)
To evaluate the integrals in Exercises 77-80. Express your answers in terms of x.
∫ arctan x dx