Q.Evaluate the following integrals:

Match each integral to its natural tool: the power rule, two substitutions, and one partial-fraction split.

(i) $\int_2^3 x^2\,dx=\left[\dfrac{x^3}{3}\right]_2^3=\dfrac{27-8}{3}=\dfrac{19}{3}.$

(ii) Let $u=30-x^{3/2}$, so $du=-\tfrac{3}{2}\sqrt{x}\,dx$, i.e. $\sqrt{x}\,dx=-\tfrac{2}{3}\,du$. Limits: $x=4\Rightarrow u=22$, $x=9\Rightarrow u=3$.

(iii) Partial fractions: $\dfrac{x}{(x+1)(x+2)}=\dfrac{-1}{x+1}+\dfrac{2}{x+2}$, so

$\left[-\log|x+1|+2\log|x+2|\right]_1^2=\log\tfrac{16}{3}-\log\tfrac{9}{2}=\log\frac{32}{27}.$ …

Each part is a proper definite integral (the integrand is finite on the whole interval), so we find an antiderivative and apply $F(b)-F(a)$. The only skill is spotting the right technique for each.

(i) $\int_2^3 x^2\,dx$

Straight power rule: $\int x^n\,dx=\dfrac{x^{n+1}}{n+1}$ with $n=2$.

(ii) $\int_4^9 \dfrac{\sqrt{x}}{(30-x^{3/2})^2}\,dx$

The derivative of the inner expression $30-x^{3/2}$ is $-\tfrac{3}{2}\sqrt{x}$ — a constant multiple of the numerator, which flags a substitution.

1. Let $u=30-x^{3/2}$. Then $du=-\tfrac{3}{2}\sqrt{x}\,dx$, so $\sqrt{x}\,dx=-\tfrac{2}{3}\,du$.
2. New limits: $x=4\Rightarrow u=30-8=22$; $x=9\Rightarrow u=30-27=3$.
3. Rewrite and integrate:

4. Simplify: $\dfrac{1}{3}-\dfrac{1}{22}=\dfrac{22-3}{66}=\dfrac{19}{66}$, so the value is $\dfrac{2}{3}\cdot\dfrac{19}{66}=\dfrac{19}{99}$.

(iii) $\int_1^2 \dfrac{x\,dx}{(x+1)(x+2)}$

A proper rational function with distinct linear factors — use partial fractions.

1. Write $\dfrac{x}{(x+1)(x+2)}=\dfrac{A}{x+1}+\dfrac{B}{x+2}$, so $x=A(x+2)+B(x+1)$.
2. Put $x=-1$: $-1=A(1)\Rightarrow A=-1$. Put $x=-2$: $-2=B(-1)\Rightarrow B=2$.
3. Integrate: $\int_1^2\left(\frac{-1}{x+1}+\frac{2}{x+2}\right)dx=\Big[-\log|x+1|+2\log|x+2|\Big]_1^2.$ …

Method: Identify-the-Technique for Each Definite Integral, Then Apply Limits

Use this for a set of definite integrals of different types: choose the antiderivative technique per integrand, then evaluate with the Fundamental Theorem, $\int_a^b f=F(b)-F(a)$.

Steps

Step 1: Classify each integrand.

Match to a technique: a plain power → power rule; a composite with its derivative present → substitution; a proper rational function → partial fractions.

Step 2: For a substitution, change the limits too. …

Common Mistakes

Mistake 1: Keeping the old $x$-limits after substituting.

Why it's wrong: once you change to $u$, the limits must become $u$-values; using the $x$-limits gives a wrong number. Correct approach: convert limits with $u=g(a)$, $u=g(b)$.

Mistake 2: Writing $+C$ in a definite integral. …