We can use the chain rule for antiderivatives to solve trigonometric functions.

Consider this indefinite integral, which includees a trigonometric expression.

After choosing an expression for u, we determine du.

Substituting u and du into the integral yields an indefinite integral that is easily solved.

We can check our answer by differentiating it.

We also can check our answer with a graph. First, we plot f(x) which represents the integrand.

Then we plot F(x) which represents a particular solution of the indefinite integral when C equals one.

Notice that f(x) is zero when F(x) is either a maximum or a minimum.

When F(x) is decreasing, f(x) is negative.

When F(x) is increasing, f(x) is positive.

These graphical results are what we expect if F(x) is the antiderivative of f(x).

Here are some antidifferentiation rules that can be used to find the antiderivative of expressions including the tangent or secant functions.

They can be used to solve indefinite integrals such as this one.

Effective strategies exist for solving indefinite integrals that include powers of sine and cosine.

If the power of sine or cosine is odd, we can use this trigonometric identity and substitute sine squared or cosine squared into the expression.

Let's see how this is done. Consider this indefinite integral.

First, we factor out one cosine factor, and then use the trigonometric identity to express the remaining cosine factors in terms of sine.

This gives us an expression with two integral terms to solve. Integrating the first term is straightforward.

After solving both integrals, we combine the terms to reach the final general solution.

If the powers of both sine and cosine are even, another strategy involving a trigonometric identity, can be used.

We'll use this strategy to solve the indefinite integral shown here.

The first step is to use this half-angle identity and make a substitution.

There is also a strategy for finding an indefinite integral that involves a power of the tangent function, such as the expression shown here.

First, we use this trigonometric identity to express tangent squared in terms of secant squared.

We then separate the integral into two integrals.

The second integral can be solved easily by using a trigonometric rule for antiderivatives.

After determining the solutions of the two integrals, we combine the terms and obtain the general solution of the original indefinite integral.

Once again, we can check our answer by differentiating it.

What should we let u equal? Click the "Submit" button after selecting your answer.

What is the general solution of this indefinite integral? Click the "Submit" button after selecting your answer.

What is the general solution of this indefinite integral? Click the "Submit" button after selecting your answer.

What is the solution of the second integral term? Click the "Submit" button after selecting your answer.

Now that we have made the substitution, what is the general solution of the resultant indefinite integral? Click the "Submit" button after selecting your answer.

What is the general solution of the first integral? Click the "Submit" button after selecting your answer.

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