Video Transcript
Determine the integral of eight times 𝑥 to the power of three 𝑒 plus seven times 𝑒 to the power of negative eight 𝑥 with respect to 𝑥.
The question wants us to determine the integral of the sum of two functions. The first thing we need to notice is that 𝑒 is just a constant. So eight times 𝑥 to the power of three 𝑒 is a polynomial term. This means we can integrate this term by using the power rule for integration as long as three times 𝑒 is not equal to negative one. Next, we see we can also integrate the second term in our integrand. Seven times 𝑒 to the power of negative eight 𝑥 is an exponential function.
So because we know how to integrate both terms inside of our integrand, we’ll split our integral of a sum of functions into the sum of two integrals. This gives us the integral of eight times 𝑥 to the power of three 𝑒 with respect to 𝑥 plus the integral of seven times 𝑒 to the power of negative eight 𝑥 with respect to 𝑥. We can now integrate both of these two terms separately. First, eight times 𝑥 to the power of three 𝑒 can be integrated by using the power rule for integration. We want to add one to our exponent of 𝑥 and then divide by this new exponent.
Finally, we need to add our constant of integration. We need to add one to our exponent of 𝑥, giving us three 𝑒 plus one and then divide by three 𝑒 plus one. Finally, we add a constant of integration we’ll call 𝑐 one. So this integral evaluates to give us eight 𝑥 to the power of three 𝑒 plus one divided by three 𝑒 plus one plus 𝑐 one. We can now evaluate our second integral. We’ll do this by using one of our standard laws for integrating exponential functions.
For constants 𝑎 and 𝑛 where 𝑛 is not equal to zero, the integral of a times 𝑒 to the power of 𝑛𝑥 with respect to 𝑥 is equal to 𝑎 times 𝑒 to the power of 𝑛𝑥 divided by 𝑛 plus the constant of integration 𝑐. We just need to divide by the coefficient of 𝑥 in our exponent. In this case, we can see the coefficient of 𝑥 in our exponent is negative eight. So we need to divide our integrand by negative eight and then add a constant of integration. This gives us seven times 𝑒 to the power of negative eight 𝑥 divided by negative eight plus the constant of integration 𝑐 two.
Finally, we’ll write seven divided by negative eight as negative seven-eighths. And we’ll combine our constants of integration, 𝑐 one and 𝑐 two, into a new constant we will call 𝑐. And this gives us our final answer. We’ve shown the integral of eight times 𝑥 to the power of three 𝑒 plus seven times 𝑒 to the power of negative eight 𝑥 with respect to 𝑥 is equal to eight times 𝑥 to the power of three 𝑒 plus one divided by three 𝑒 plus one minus seven over eight times 𝑒 to the power of negative eight 𝑥 plus 𝑐.
