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A short H2 Maths revision video on H2 Maths 5.3 - Integration by Parts: ∫ x² ln x dx, built for quick recap before tutorial practice or exam revision.
Read through the explanation after watching, or jump straight to the step you want to replay.
Step 1 - State the problem and recall the formula
We want to find the integral of x squared times the natural log of x.
Step 1 - State the problem and recall the formula
Integration by parts says the integral of u d v equals u v minus the integral of v d u.
Step 1 - State the problem and recall the formula
The key decision is which factor to call u and which to call d v.
Step 2 - Choose u and dv using LIATE
Use the LIATE rule: Logs, Inverse trig, Algebra, Trig, Exponentials.
Step 2 - Choose u and dv using LIATE
The factor closer to the L end becomes u, because its derivative simplifies.
Step 2 - Choose u and dv using LIATE
Logarithm beats algebra, so let u equal ln x and d v equal x squared d x.
Step 2 - Choose u and dv using LIATE
If you chose u equals x squared instead, you would need the integral of ln x, which is harder.
Step 3 - Compute du and v
Differentiate u: the derivative of ln x is one over x, so d u equals one over x d x.
Step 3 - Compute du and v
Integrate d v: the integral of x squared is x cubed over three, so v equals x cubed over three.
Step 4 - Substitute into the formula
Substitute into integration by parts.
Step 4 - Substitute into the formula
u v gives x cubed over three times ln x.
Step 4 - Substitute into the formula
Minus the integral of v d u gives minus the integral of x cubed over three times one over x d x.
Step 4 - Substitute into the formula
The x cubed over x simplifies to x squared, so we just need the integral of x squared over three.
Step 5 - Evaluate and state the final answer
The integral of x squared is x cubed over three, so we get x cubed over 3 ln x minus x cubed over 9 plus C.
Step 5 - Evaluate and state the final answer
Factor out x cubed over nine to get the neatest form: x cubed over nine times three ln x minus one, plus C.
Step 5 - Evaluate and state the final answer
Common mistake: forgetting the minus sign in front of the second integral. Always write out u v minus the integral of v d u in full before simplifying.