The Indefinite Integral Help

Example 1

Calculate

∫ sin(3x + 1)dx .

Solution 1

We know that we must guess a trigonometric function. Running through the choices, cosine seems like the best candidate. The derivative of cos x is − sin x. So we immediately see that − cos x is a better guess—its derivative is sin x . But then we adjust our guess to F (x) = − cos(3x + 1) to take into account the form of the argument. This almost works: we may calculate that F′'(x) = 3 sin(3x + 1). We determine that we must adjust by a factor of 1/3. Now we can record our final answer as

We invite the reader to verify that the derivative of the answer on the right-hand side gives sin(3x + 1).

Example 2

Calculate

Solution 2

We notice that the numerator of the fraction is nearly the derivative of the denominator. Put in other words, if we were asked to integrate

then we would see that we are integrating an expression of the form

(which we in fact encountered among our differentiation rules in Section 2.5). As we know, expressions like this arise from differentiating log (x).

Returning to the original problem, we pose our initial guess as log[x² + 3]. Differentiation of this expression gives the answer 2x /[ x² + 3]. This is close to what we want, but we must adjust by a factor of 1/2. We write our final answer as

Calculate the indefinite integral

Solution 3

We observe that the expression 6x² + 4x is nearly the derivative of x³ + x² + 1. In fact if we set (x) = x³ + x² + 1 then the integrand (the quantity that we are asked to integrate) is

[ (x)]⁵⁰ · 2 ′ (x).

It is natural to guess as our antiderivative [ (x)]⁵. Checking our work, we find that

([ (x)]⁵¹)′ = 51[ (x)]⁵⁰ · ′ (x).

We see that the answer obtained is quite close to the answer we seek; it is off by a numerical factor of 2/51. With this knowledge, we write our final answer as