Unit 10: Integration by Parts for Definite Integrals

Having introduced the integration by parts technique for indefinite integrals in Unit 5, let’s now extend this method to definite integrals. When applying integration by parts to a definite integral, we use the same formula, but with limits of integration included:

[latex]\int_a^b u dv = [uv]_a^b - \int_a^b v du[/latex]

Here’s the procedure:

1. Choose [latex]u[/latex] and [latex]dv[/latex], differentiating [latex]u[/latex] to find [latex]du[/latex] and integrating [latex]dv[/latex] to find [latex]v[/latex]. This step is same as in Unit 5.
2. Evaluate [latex]uv[/latex] at the boundaries [latex]a[/latex] and [latex]b[/latex], i.e.  [latex][uv]_a^b=u(b)v(b)-u(a)v(a)[/latex]
3. Integrate the remaining integral [latex]\int_a^b v du[/latex] within the limits [latex]a[/latex] and [latex]b[/latex].

Q3. [latex]\int_{-1}^{3} x^2e^x\;dx[/latex]    (similar to Q3 in Unit 5)

Q4. [latex]\int_{-\pi/4}^{\pi/2} \cos^3 x \;dx[/latex]    (similar to Q5 in Unit 5)

Q5. [latex]\int_1^7  t \sqrt{t+2} \;dt[/latex]    (similar to Q6 in Unit 5)