Integrate the following functions using the appropriate rules of integration. a. \( \int_{1}^{2} x \ln (x) d x \) (5 Marks) b. \( \int_{0}^{2} x\left(x^{2}-1\right)^{7} d x \) (7 Marks)

To solve the integrals, we'll proceed step by step. a. We want to calculate \( \int_{1}^{2} x \ln(x) \, dx \). To do this, we can use integration by parts, which states: \[ \int u \, dv = uv - \int v \, du \] Let \( u = \ln(x) \) (thus \( du = \frac{1}{x} \, dx \)) and \( dv = x \, dx \) (therefore \( v = \frac{x^2}{2} \)). Applying integration by parts: \[ \int x \ln(x) \, dx = \frac{x^2}{2} \ln(x) - \int \frac{x^2}{2} \cdot \frac{1}{x} \, dx = \frac{x^2}{2} \ln(x) - \frac{1}{2} \int x \, dx \] Now, evaluating the integral \( \int x \, dx = \frac{x^2}{2} \) yields: \[ \int x \ln(x) \, dx = \frac{x^2}{2} \ln(x) - \frac{1}{4} x^2 + C \] Now we need to evaluate from 1 to 2: \[ \left. \left( \frac{x^2}{2} \ln(x) - \frac{1}{4} x^2 \right) \right|_{1}^{2} \] Substituting the bounds gives us: At \( x = 2 \): \[ \frac{2^2}{2} \ln(2) - \frac{1}{4}(2^2) = 2 \ln(2) - 1 \] At \( x = 1 \): \[ \frac{1^2}{2} \ln(1) - \frac{1}{4}(1^2) = 0 - \frac{1}{4} = -\frac{1}{4} \] Thus, we combine these results: \[ \left( 2 \ln(2) - 1 \right) - \left( -\frac{1}{4} \right) = 2 \ln(2) - 1 + \frac{1}{4} = 2 \ln(2) - \frac{4}{4} + \frac{1}{4} = 2 \ln(2) - \frac{3}{4} \] So, the value of the integral is: \[ \int_{1}^{2} x \ln(x) \, dx = 2 \ln(2) - \frac{3}{4}. \] b. Now for the integral \( \int_{0}^{2} x (x^2 - 1)^7 \, dx \). Here, we can also use substitution. Let \( u = x^2 - 1 \). Then, \( du = 2x \, dx \), or \( dx = \frac{du}{2x} \), which means that \( x = \sqrt{u + 1} \). Now, adjusting the limits: When \( x = 0 \), \( u = 0^2 - 1 = -1 \). When \( x = 2 \), \( u = 2^2 - 1 = 3 \). Now substituting: \[ \int_{0}^{2} x (x^2 - 1)^7 \, dx = \frac{1}{2} \int_{-1}^{3} (u)^{7} \, du \] The integral of \( u^7 \) is: \[ \frac{u^8}{8} \] So we evaluate: \[ \frac{1}{2} \left[ \frac{u^8}{8} \right]_{-1}^{3} = \frac{1}{2} \left[ \frac{3^8}{8} - \frac{(-1)^8}{8} \right] = \frac{1}{2} \left[ \frac{6561}{8} - \frac{1}{8} \right] = \frac{1}{2} \cdot \frac{6560}{8} = \frac{6560}{16} = 410 \] Thus, the integral is: \[ \int_{0}^{2} x (x^2 - 1)^7 \, dx = 410. \]