# Video: Finding the Equation of a Curve given the Slope of Its Tangent by Using Indefinite Integration

The slope at the point (π₯, π¦) on the graph of a function is dπ¦/dπ₯ = β4π sin ππ₯ + 5π cos ππ₯. Find the equation of the curve if it contains the point (1, 2).

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### Video Transcript

The slope at the point π₯, π¦ on a graph of a function is dπ¦ by dπ₯ equals negative four π sin of ππ₯ plus five π cos of ππ₯. Find the equation of the curve if it contains the point one, two.

In this question, weβve been given information about the derivative of our original function. Now, weβre looking to find the equation of the curve, given some information about a point that it passes through. In other words, we need to establish an equation for π¦ in terms of π₯. And so we begin by recalling that the fundamental theorem of calculus tells us that integration and differentiation are essentially reverse processes. And so weβll find an equation for π¦ by integrating our expression for dπ¦ by dπ₯ with respect to π₯.

In this case, π¦ is then the indefinite integral of negative four π sin of ππ₯ plus five π cos of ππ₯ with respect to π₯. And actually, we can quote the general result for the integral of the sine and cosine functions. The integral of sin of ππ₯ for some real constants π is negative one over π cos of ππ₯ plus a constant of integration π. Similarly, integrating cos of ππ₯ with respect to π₯ is one over π times sin of ππ₯ plus π. Remember, we can integrate term by term. So we begin by integrating negative four π of sin ππ₯ with respect to π₯. Itβs negative four π times negative one over π cos of ππ₯, which is four cos of ππ₯.

Then when we integrate five π cos of ππ₯, we get five π times one over π sin of ππ₯, which is five sin of ππ₯. And since weβre performing an indefinite integral, itβs really important that we have that constant of integration π. So weβve found that π¦ is equal to four cos of ππ₯ plus five sin of ππ₯ plus π. Now, we really need to find the value of π. So we go back to the question and the information that the curve passes through the point one, two. In other words, when π₯ is equal to one, π¦ is equal to two. And we substitute these values into our equation. We get two equals four cos of π times one plus five sin of π times one plus π.

Now, we know that five sin of π is zero. Similarly, cos of π is negative one. So four cos of π is negative four. And our equation becomes two equals negative four plus π, which we can solve for π by adding four to both sides. So π is equal to six. And we can, therefore, replace π with six in our equation for π¦. Thatβs π¦ equals four cos of ππ₯ plus five sin of ππ₯ plus six. Or if we choose by convention to write summing first, we get π¦ equals five sin of ππ₯ plus four cos of ππ₯ plus six.