# Integration by Parts

Mathematics I · Integral Calculus · https://tryals.app/learn/mathematics-i/integration-by-parts

## Reversing the Product Rule

**Integration by parts** reverses the **product rule** to handle integrals of products that substitution cannot touch. The formula is:
$$\int u\, dv = uv - \int v\, du$$
Here, $u$ is differentiated and $dv$ is integrated, trading your original integral for a hopefully simpler one.

The art is **choosing $u$ and $dv$ wisely**. Use the mnemonic **LIATE** to order your choices for $u$:

| Priority | Category | Example |
| --- | --- | --- |
| 1 | **L**ogarithmic | $\ln(x)$ |
| 2 | **I**nverse trig | $\arctan(x)$ |
| 3 | **A**lgebraic | $x^2$ |
| 4 | **T**rigonometric | $\sin(x)$ |
| 5 | **E**xponential | $e^x$ |

Pick $u$ as the type appearing earliest in LIATE, and let $dv$ contain the rest.

## Worked Example & Pitfalls

For $\int x e^x\, dx$, choose **$u = x$** (algebraic, giving $du = dx$) and **$dv = e^x\, dx$** (giving $v = e^x$). Applying the formula:
$$\int x e^x\, dx = x e^x - \int e^x\, dx = x e^x - e^x + C$$
Sometimes parts must be applied **twice**, like for $\int x^2 e^x\, dx$.

> **Common pitfall:** choosing $u$ and $dv$ **badly**, making $\int v\, du$ harder than the original. Always pick $u$ to get _simpler_ when differentiated.

Recall the core split:
- Substitution is for **compositions**.
- Parts is for **products**.

## Practice questions

5 of this lesson's 11 practice questions, with answers. The full set is in the app.

### 1. Select every integral that integration by parts handles well rather than substitution.

A. (x^2 + 1)^5 times 2x dx
B. x^2 e^x dx
C. x sin x dx
D. ln x dx
E. 2x cos(x^2) dx

**Answer:** B. x^2 e^x dx; C. x sin x dx; D. ln x dx

**Why:** Options 3 and 5 are pure substitutions ($u = x^{2}$ and $u = x^{2}+1$, with the inner derivative present). The other three are products of unrelated types — including the lone ln x, taken with dv = dx — and belong to parts.

Page: https://tryals.app/practice/mathematics-i/integration-by-parts/select-every-integral-that-integration-by-parts-handles-well-rather

### 2. Integration by parts turns ∫u dv into uv − ∫v du - trading one integral for another. Why is that progress rather than circular?

A. Because the new integral is always simpler, whatever choice of u and dv is made.
B. Because the formula eliminates the product, and integrals of single functions are always elementary.
C. Because uv is always the answer, and the remaining integral vanishes.
D. Because u and dv are chosen so the new integral is easier.

**Answer:** D. Because u and dv are chosen so the new integral is easier.

**Why:** With u = x and dv = eˣ dx, differentiating x turns it into 1 and the remaining integral is trivial, that is the progress. The wrong choice shows it is not automatic: u = eˣ raises the polynomial's degree and leaves a harder integral. LIATE encodes which factors improve on differentiation, and exponentials belong at the dv end because they never simplify.

Page: https://tryals.app/practice/mathematics-i/integration-by-parts/integration-by-parts-turns-u-dv-into-uv-v-du-trading-one

### 3. Order the steps of integration by parts.

**Answer:**

1. Recognise the integrand as a product of two different types of function
2. Choose u by LIATE and let dv be everything that is left, including dx
3. Differentiate u to get du and integrate dv to get v
4. Write down u v minus the integral of v du
5. Evaluate the new integral, applying parts again if a polynomial factor remains

**Why:** Parts is bookkeeping: split, differentiate one half, integrate the other, assemble. Repeating the process is normal, each pass lowers the degree of a polynomial factor by one, so $x^{3}e^{x}$ needs three passes.

Page: https://tryals.app/practice/mathematics-i/integration-by-parts/order-the-steps-of-integration-by-parts

### 4. Sort each integral by which factor LIATE tells you to take as u.

**Answer:**

- Take u = the polynomial part: x e^x dx, x cos x dx, x^2 e^x dx
- Take u = the logarithm: x ln x dx, x^2 ln x dx

**Why:** Whenever a logarithm is present it wins, because ln x has no elementary antiderivative you would want as v but differentiates to a tidy 1/x. Against exponentials and trig functions the polynomial wins, since differentiating it drives the degree down to zero.

Page: https://tryals.app/practice/mathematics-i/integration-by-parts/sort-each-integral-by-which-factor-liate-tells-you-to-take-as-u

### 5. Integrating ∫x·eˣ dx by parts works; ∫eˣ·x dx with the choices reversed produces a harder integral. The integrand is identical. What does that show about the technique?

A. That the second attempt contains an arithmetic error
B. That the integrand must be written in a specific order for the rule to apply
C. That it is a tool requiring judgement rather than a procedure
D. That integration by parts fails for products of polynomials and exponentials

**Answer:** C. That it is a tool requiring judgement rather than a procedure

**Why:** Both applications are valid; only one makes progress, because differentiating x simplifies it and differentiating eˣ does not. LIATE is a heuristic encoding that observation, and it is why integration is a craft where differentiation is an algorithm.

Page: https://tryals.app/practice/mathematics-i/integration-by-parts/integrating-x-e-dx-by-parts-works-e-x-dx-with-the-choices