# Integration by Substitution

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

## Reversing the Chain Rule

**Integration by substitution** is the reverse of the chain rule and your most powerful tool for solving complex integrals. It transforms a hard integral into an easy one through a clever change of variable.

When your integrand contains an inner function $u = g(x)$ alongside its derivative factor, apply the formal rule:

$$\int f(g(x))\,g'(x)\, dx = \int f(u)\, du$$

Here, $u = g(x)$ and $du = g'(x)\, dx$. This collapses the composite expression into a simple integral in $u$ which you evaluate before converting back.

| Step | Action | Description |
| --- | --- | --- |
| **1** | Choose $u$ | Pick the inner function whose derivative also appears. |
| **2** | Compute $du$ | Find $du = g'(x)\, dx$ and rewrite the integral entirely in $u$. |
| **3** | Integrate | Solve in terms of $u$, then substitute $x$ back in. |

## Worked Example & Pitfalls

Evaluate $\int 2x\cos(x^2)\, dx$. Let **$u = x^2$**, making **$du = 2x\, dx$**. Substituting these gives $\int \cos u\, du = \sin u + C = \sin(x^2) + C$.

For **definite integrals**, you have two valid paths:

| Method | Strategy | Advantage |
| --- | --- | --- |
| **Substitute back** | Solve the $u$-integral, convert back to $x$, then use original limits. | Familiar process |
| **Change limits** | Convert $x$-limits via $u = g(x)$ and evaluate directly in $u$. | Faster, no back-substitution |

> **Common Pitfalls:** Never forget to convert $dx$ into $du$ by accounting for the $g'(x)$ factor. In definite integrals, never mix old $x$-limits with the new $u$-variable; change your limits immediately or substitute back first.

## Practice questions

5 of this lesson's 10 practice questions, with answers. The full set is in the app.

### 1. Select every integral for which u = x^3 + 2 is a productive substitution.

A. 3x^2 (x^3 + 2)^5 dx
B. x^2 times the square root of (x^3 + 2) dx
C. x^2 e^(x^3 + 2) dx
D. (x^3 + 2)^5 dx
E. 3x (x^3 + 2) dx

**Answer:** A. 3x^2 (x^3 + 2)^5 dx; B. x^2 times the square root of (x^3 + 2) dx; C. x^2 e^(x^3 + 2) dx

**Why:** Options 1, 2 and 4 all carry an $x^{2}$ factor, which is $3x^{2}$ up to the constant 1/3, so the substitution closes. Option 3 has no such factor at all, and option 5 offers x rather than $x^{2}$, a missing _variable_ factor cannot be fixed by pulling out a constant.

Page: https://tryals.app/practice/mathematics-i/integration-by-substitution/select-every-integral-for-which-u-x-3-2-is-a-productive

### 2. Substitution fails on $\int \cos(x^{2})\,dx$ because the derivative of the inner function is not present as a factor.

**Answer:** True

**Why:** True. Substitution undoes the chain rule, so it needs the inner function's derivative sitting there as a factor, the $2x$ that $\int 2x\cos(x^{2})dx$ has and $\int\cos(x^{2})dx$ does not. The second has no elementary antiderivative at all; the Fresnel integral is defined to name it.

Page: https://tryals.app/practice/mathematics-i/integration-by-substitution/substitution-fails-on-cos-x-dx-because-the-derivative-of-the

### 3. Order the steps of integration by substitution for an indefinite integral.

**Answer:**

1. Spot an inner function whose derivative also appears in the integrand
2. Set u equal to that inner function
3. Differentiate to get du = g'(x) dx and rewrite every x-part in terms of u
4. Integrate the resulting simpler integral in u
5. Substitute u = g(x) back so the answer is a function of x

**Why:** The order matters because the conversion of dx into du is what makes the method valid, not a formality. For a definite integral you may replace the final step by converting the limits instead.

Page: https://tryals.app/practice/mathematics-i/integration-by-substitution/order-the-steps-of-integration-by-substitution-for-an-indefinite

### 4. Substituting $u = x^{2}$ into $\int 2x\cos(x^{2})\,dx$ gives $\int \cos u\,du$, not $\int \cos u\,dx$. Why must the differential change too?

A. Because $dx$ may be left unchanged whenever the substitution is linear
B. Because du = 2x dx replaces both the factor and the dx
C. Because $dx$ and $du$ are equal for this substitution
D. Because the constant of integration absorbs the difference

**Answer:** B. Because du = 2x dx replaces both the factor and the dx

**Why:** Substitution is a change of variable throughout, integrand and differential together. The $2x$ is consumed precisely by converting $dx$ into $du$.

Page: https://tryals.app/practice/mathematics-i/integration-by-substitution/substituting-u-x-into-2xcos-x-dx-gives-cos-u-du-not-cos-u

### 5. Substitution requires changing the limits too when working with a definite integral. What goes wrong if you keep the original ones?

A. The integral would be negated, since substitution reverses orientation
B. You would evaluate at the old variable's limits, which belong to x
C. The answer would be correct but the working invalid
D. Nothing, provided you substitute back to x before evaluating

**Answer:** B. You would evaluate at the old variable's limits, which belong to x

**Why:** Limits are values of the variable, so changing the variable changes them: x from 0 to 2 becomes u from 0 to 4. Option 2 names the legitimate alternative, substitute back and keep the old limits, and the error is doing neither, which mixes u's function with x's endpoints.

Page: https://tryals.app/practice/mathematics-i/integration-by-substitution/substitution-requires-changing-the-limits-too-when-working-with-a