Integration by parts | Year 12

01 / LEARN

The idea, step by step

Reverse the product rule

Integrating the product-rule identity gives $\int u d v = u v - \int v d u$ . Choose u so its derivative simplifies, and choose dv so it has a manageable antiderivative. Keep the minus sign attached to the whole remaining integral.

Polynomials paired with exponentials or trigonometric functions often simplify when the polynomial is differentiated. For logarithms or inverse trigonometric functions alone, use an implicit factor 1 as dv.

Repeat and verify

A quadratic polynomial times an exponential usually requires two applications. Each stage should reduce the polynomial degree. Some cyclic products return the original integral; collect it algebraically rather than continuing forever.

For definite integrals, apply bounds to both the uv term and the remaining integral, or find a full antiderivative first. Differentiate your final result to check product signs and inner-frequency factors.

A common trap

The product antiderivative is the product of the two antiderivatives.

Instead: Use the product-rule identity; simple multiplication generally differentiates to extra terms.

02 / WORKED EXAMPLES

See the thinking

Example 1

Worked example 1

Find $\int x e^{x} d x$ .

Choose u=x and dv=exp(x)dx.
Then the result is x exp(x) minus exp(x).

(x - 1) e^{x} + C

Example 2

Worked example 2

Find $\int ln x d x$ for x>0.

Choose u=ln x, dv=dx.
Then $u v - \int v d u = x ln x - \int 1 d x$ .

x ln x - x + C

Check your understanding

For $\int x cos x d x$ , choose u and dv.

Show answer

$u = x$ , $d v = cos x d x$ .

Differentiating x simplifies it, while cosine is easy to integrate.

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Find $\int x cos x d x$ .
2 marks
Find $\int x sin 2 x d x$ .
2 marks
Find $\int x^{2} e^{x} d x$ .
2 marks
Find $\int x ln x d x$ for x>0.
2 marks
Evaluate $\int_{0}^{1} x e^{x} d x$ .
2 marks
Find $\int arctan x d x$ .
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Find $\int x cos x d x$ .
2 marks
Use u=x, v=sin x.
Subtract the integral of sin x.
$x sin x + cos x + C$ .
Find $\int x sin 2 x d x$ .
2 marks
Use v=-cos2x/2.
Integrate the remaining cosine term.
$- x cos 2 x / 2 + sin 2 x / 4 + C$ .
Find $\int x^{2} e^{x} d x$ .
2 marks
Apply parts with u=x squared.
Apply parts again to the remaining x exp(x) term.
$e^{x} (x^{2} - 2 x + 2) + C$ .
Find $\int x ln x d x$ for x>0.
2 marks
Use u=ln x, dv=x dx.
The remaining integral is x/2.
$(x^{2} / 2) ln x - x^{2} / 4 + C$ .
Evaluate $\int_{0}^{1} x e^{x} d x$ .
2 marks
Use antiderivative (x-1)exp(x).
The endpoint values are 0 and -1.
$1$ .
Find $\int arctan x d x$ .
2 marks
Use u=arctan x, dv=dx.
The remaining integral is x/(1+x squared).
$x arctan x - \frac{1}{2} ln (1 + x^{2}) + C$ .

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 2 Specialist Mathematics integration techniques and applications, including inverse-trigonometric patterns, substitution, parts, simple partial fractions, areas and solids of revolution.

Official curriculum source ↗

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