Question Video: Integration of Definite Integrals Mathematics

Video Transcript

Evaluate the integral from one to two of eight 𝑡 divided by 10𝑡 squared plus five with respect to 𝑡.

We’re asked to calculate the integral of a rational function, and we’ve seen many methods for doing this. We need to decide which one we’re going to apply in this case. We can notice the derivative of our denominator 10𝑡 squared plus five with respect to 𝑡 is equal to 20𝑡. And we see this is a scalar multiple of our numerator. Then if we set our denominator equal to the function 𝑔 of 𝑡, we can notice we’re asked to integrate a scalar multiple 𝑘 of the derivative of 𝑔 divided by 𝑔 of 𝑡 with respect to 𝑡. In fact, this is a special case of an integration by substitution rule we’ve seen before when we set our function 𝑓 of 𝑡 equal to one divided by 𝑡.

We recall if our function 𝑔 prime is continuous on the closed interval from 𝑎 to 𝑏 and our function 𝑓 is continuous on the range of 𝑢 is equal to 𝑔 of 𝑡. Then by using integration by substitution with the substitution 𝑢 is equal to 𝑔 of 𝑡. We have the integral from 𝑎 to 𝑏 of 𝑔 prime of 𝑡 multiplied by 𝑓 evaluated at 𝑔 of 𝑡 with respect to 𝑡 is equal to the integral from 𝑔 evaluated at 𝑎 to 𝑔 evaluated at 𝑏 of 𝑓 evaluated at 𝑢 with respect to 𝑢. And we’ve already shown we can write our integral in this manner. In fact, since 𝑘 is a constant, we can take it outside of our integral.

So we want to use the substitution 𝑢 is equal to 𝑔 of 𝑡 which is 10𝑡 squared plus five. This gives us that our function 𝑔 prime of 𝑡 is equal to 20𝑡 which is continuous on all of the real numbers, which means, in particular, it must be continuous on the closed interval of our integral, which in this case is the closed interval from one to two. Next, we need to check that our function 𝑓, which in this case is one divided by 𝑡, is continuous on the range of 𝑢 is equal to 𝑔 of 𝑡. We find the range of 𝑔 of 𝑡 by evaluating the function at the limits of our integral. That’s 𝑔 of two and 𝑔 of one.

We have that 𝑔 evaluated at two is equal to 10 multiplied by two squared plus five, which we can evaluate to give us 45. Similarly, we can evaluate 𝑔 at one to give us 10 multiplied by one squared plus five, which is equal to 15. So we need our function 𝑓 to be continuous on the closed interval from 15 to 45. And since our function 𝑓 of 𝑡 is just equal to one divided by 𝑡 is continuous on any interval which does not include zero, so, in particular, it’s continuous on the range of 𝑔 of 𝑡. So we’re now ready to use integration by substitution by using 𝑢 is equal to 10𝑡 squared plus five.

Differentiating both sides of this equation gives us that d𝑢 d𝑡 is equal to 20𝑡. And although d𝑢 by d𝑡 is not a fraction, when using integration by substitution, it does behave a little bit like a fraction. This gives us the equivalent statement d𝑢 is equal to 20𝑡 d𝑡. We’re almost ready to evaluate this integral. We just need to rewrite our numerator as a scalar multiple of the derivative of 𝑢. And we see if our constant 𝑘 is equal to eight divided by 20, then eight divided by 20 multiplied by 20𝑡 is equal to eight 𝑡. And we can take this constant of eight divided by 20 outside of our integral. We now have that our denominator of 10𝑡 squared plus five is equal to 𝑢 and 20𝑡 d𝑡 is equal to d𝑢.

Finally, the new limits of our integral will be from 𝑔 evaluated at one to 𝑔 evaluated at two. That’s from 15 to 45. So the substitution 𝑢 is equal to 10𝑡 squared plus five gave us that our integral is equal to eight divided by 20 multiplied by the integral from 15 to 45 of one divided by 𝑢 with respect to 𝑢. Now, we can evaluate our integral by using the fact that the integral of one divided by 𝑥 with respect to 𝑥 is equal to the natural logarithm of the absolute value of 𝑥 plus a constant of integration 𝐶. We simplify eight divided by 20 to give us two divided by five. Then we integrate one divided by 𝑢 to give us the natural logarithm of the absolute value of 𝑢, while we ignore our constant of integration because it will cancel when we evaluate at the limits of our integral 𝑢 is equal to 15 and 𝑢 is equal to 45.

Evaluating at the limits of our integral gives us two-fifths multiplied by the natural logarithm of the absolute value of 45 minus the natural logarithm of the absolute value of 15. Both 45 and 15 are positive. So their absolute values are just equal to themselves. So we can remove the absolute value. Finally, we can simplify this by using the quotient rule for logarithms, which tells us the log of 𝑎 minus the log of 𝑏 is equal to the log of 𝑎 divided by 𝑏. The difference between two logarithms is equal to the quotient of the logarithm.

So the natural logarithm of 45 minus the natural logarithm of 15 can be simplified to the natural logarithm of 45 divided by 15. And 45 over 15 simplifies to three, giving us that our integral is two times the natural logarithm of three divided by five. Therefore, by using integration by substitution, we’ve shown that the integral from one to two of eight 𝑡 divided by 10𝑡 squared plus five with respect to 𝑡 is equal to two times the natural logarithm of three all divided by five.