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Evaluate the integrals by making appropriate u-substitutions and applying the formulas reviewed in this section.$$\int 4 x \tan \left(x^{2}\right) d x$$

Calculus 1 / AB

Calculus 2 / BC

Chapter 7

PRINCIPLES OF INTEGRAL EVALUATION

Section 1

An Overview of Integration Methods

Integrals

Integration

Integration Techniques

Trig Integrals

Trig Substitution

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Lectures

02:15

In mathematics, a trigonom…

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In mathematics, trigonomet…

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Evaluate the integrals by …

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Evaluate the integrals.

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Evaluate the integrals…

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Evaluate the integrals usi…

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eso in this question, we're going to be solving for the integral using the method of you substitution. So we have the integral of four x of times tan of X squared times the X and so first thing we're going to do is we're going to set Are you equal to X squared? And now we're going to take the derivative of both sides. So we're going to have d u is equal to two x d x. So d you over to X is equal to d X And now from here we're going to do is we're gonna replace our X squared with Are you and r d acts with r D you over to acts. So now we're gonna have the integral of four x times tan of the U. D you over two x and our exes will cross out completely and we can simplify Um R four and r two or four is the new divide By two. We're left with two. We're gonna have the integral of to tan you do you and we can take out our to and which is going to be with e left with the tan of you d you and the tan of you. The integral of that is just equal to the negative natural log of the coat sign of you plus C. And this is times are so we're going tohave negative two times the l end of coasts, you plus C. And the last thing we need to do is replace. Are you with our X squared? Because are you is equal to X squared. So we're going to have as our final answer negative two times the natural log of the coasts of X squared plus C.

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