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VICMath MethodsQuick questions
Unit 4
Quick questions on Integration by substitution: VCE Math Methods Unit 4
14short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.
What is example 1. Power inside a polynomial?Show answer
Choose $u = x^2 + 1$. Then $du = 2x \, dx$, so $6 x \, dx = 3 \, du$.
What is example 2. Exponential of a function?Show answer
Choose $u = x^2$. Then $du = 2x \, dx$, so $x \, dx = \frac{1}{2} du$.
What is example 3. $\frac{1}{x}$-style logarithmic?Show answer
$\int \frac{2x + 3}{x^2 + 3x + 5} \, dx$.
What is example 4. Trigonometric?Show answer
$\int \sin^3(x) \cos(x) \, dx$.
What is example. Option A?Show answer
$\int_{0}^{1} 2x (x^2 + 1)^3 \, dx$.
What is example. Option B (same integral)?Show answer
Compute antiderivative in $u$. $\frac{u^4}{4} = \frac{(x^2 + 1)^4}{4}$.
What is option A. Change the limits?Show answer
When $u = g(x)$, the lower limit $x = a$ becomes $u = g(a)$ and the upper limit $x = b$ becomes $u = g(b)$. Evaluate the integral in $u$ directly between the new limits. No need to substitute back.
What is option B. Substitute back?Show answer
Compute the antiderivative in $u$, replace $u$ with $g(x)$, then evaluate at the original $x$-limits.
What is choosing $u$ as the wrong piece?Show answer
The standard heuristic: $u$ is the inside function whose derivative appears in the integrand. Choosing $u = x$ or $u = $ the outside function rarely simplifies.
What is forgetting to substitute $du$ for $dx$?Show answer
$\int f(u) \cdot dx$ is mixed notation; the $dx$ must be converted to $du$ before integrating.
What is forgetting to change the limits in definite integrals?Show answer
If you choose Option A but use the original $x$-limits with the $u$-integrand, the arithmetic is wrong.
What is not substituting back in indefinite integrals?Show answer
$\frac{u^5}{5} + c$ is not a complete answer for an integral originally in $x$; replace $u$ with $g(x)$ before submitting.
What is constants getting lost?Show answer
If the derivative of the inside is $2x$ but only $x$ is in the integrand, you must compensate with a factor of $\frac{1}{2}$. Forgetting halves the answer.
What is using substitution when the linear reverse chain rule would do?Show answer
For $\int (3x + 1)^4 \, dx$, the reverse chain gives $\frac{(3x + 1)^5}{15} + c$ directly; full substitution works but is slower.