Evaluating the Integral: ∫₁⁴ 3√t ln(t) dt

In this article, we will learn how to evaluate the definite integral of the function 3√t ln(t) over the interval [1, 4].

The integral we need to evaluate is:

∫₁⁴ 3√t ln(t) dt

To evaluate this integral, we will use the fundamental theorem of calculus and the properties of integrals.

Step 1: Understand the function

The function we are integrating is 3√t ln(t), which is the product of two functions:

• 3√t: the square root of t, raised to the power of 3
• ln(t): the natural logarithm of t

Step 2: Apply the power rule and logarithm rule

To evaluate the integral, we can use the power rule and the logarithm rule of integration:

∫ x^n ln(x) dx = (x^(n+1) / (n+1)) ln(x) - ∫ x^(n+1) / (n+1) dx

In our case, n = 3/2, so we can apply the formula:

∫ 3√t ln(t) dt = (t^(3/2+1) / (3/2+1)) ln(t) - ∫ t^(3/2+1) / (3/2+1) dt

Simplifying the expression, we get:

∫ 3√t ln(t) dt = (2t^(5/2) / 5) ln(t) - ∫ 2t^(5/2) / 5 dt

Step 3: Evaluate the integral

Now, we can evaluate the integral from 1 to 4:

∫₁⁴ 3√t ln(t) dt = [(2t^(5/2) / 5) ln(t)]₁⁴ - ∫₁⁴ 2t^(5/2) / 5 dt

Evaluating the first term:

(2t^(5/2) / 5) ln(t)
= (2 * 4^(5/2) / 5) ln(4) - (2 * 1^(5/2) / 5) ln(1)
= (2 * 32 / 5) ln(4) - 0
= 12.8 ln(4)

Evaluating the second term:

∫₁⁴ 2t^(5/2) / 5 dt
= (2 / 5) ∫₁⁴ t^(5/2) dt
= (2 / 5) [(2t^(7/2) / 7)]₁⁴
= (2 / 5) [(2 * 4^(7/2) / 7) - (2 * 1^(7/2) / 7)]
= (2 / 5) [(128 / 7) - (2 / 7)]
= 36.57

Combining the two terms, we get the final result:

∫₁⁴ 3√t ln(t) dt = 12.8 ln(4) - 36.57 = -23.77

Therefore, the value of the definite integral ∫₁⁴ 3√t ln(t) dt is -23.77.