Substitution and Integral Structure

The integral of $2x\cos(x^2)$ contains a composition and the derivative of its inner function. The chain rule says $d[\sin(x^2)]/dx=2x\cos(x^2)$, so reversing it supplies an antiderivative. Substitution organizes that recognition.

The theorem behind the notation

If $F'=f$, then the chain rule gives $(F\circ g)'=f(g)g'$. Consequently,

$$\int f(g(x))g'(x)\,dx=F(g(x))+C.$$

Under continuous integrand and continuously differentiable substitution hypotheses on $[a,b]$, FTC yields

$$\int_a^bf(g(x))g'(x)\,dx =\int_{g(a)}^{g(b)}f(u)\,du.$$

The substitution need not be one-to-one for this signed one-dimensional identity when these conditions hold: the chain-rule proof handles reversals automatically. More general change-of-variable theorems require their own conditions. The shorthand $u=g(x)$ and $du=g'(x)dx$ records this theorem; it is not permission to replace pieces of an integrand inconsistently.

Matching the differential factor

For $\int x\sqrt{1+x^2}\,dx$, let $u=1+x^2$, giving $du=2x\,dx$. Then

$$\int x\sqrt{1+x^2}\,dx =\frac12\int u^{1/2}\,du =\frac13(1+x^2)^{3/2}+C.$$

Differentiate the result: $(1/3)(3/2)(1+x^2)^{1/2}(2x)=x\sqrt{1+x^2}$. This verifies the coefficient, the most common source of a substitution error.

For $\int 3x^2/(1+x^3)\,dx$, use $u=1+x^3$, obtaining $\log|1+x^3|+C$ on intervals avoiding $x=-1$. The absolute value and domain restriction remain necessary.

Definite bounds belong to the new variable

Evaluate $\int_0^1 2x e^{x^2}\,dx$. With $u=x^2$, the bounds transform from $0,1$ to $0,1$, so the result is $\int_0^1e^u\,du=e-1$. For $\int_1^2 2x e^{x^2}\,dx$, the transformed bounds are $1,4$, yielding $e^4-e$. The first example happens to preserve the numerical bounds; that coincidence is not a rule.

One may instead find an antiderivative in $x$ and then use the original bounds. What fails is evaluating a $u$-antiderivative at bounds that still represent $x$ without transformation.

For $\int_0^{\pi/2}\sin x\cos x\,dx$, let $u=\cos x$, $du=-\sin x\,dx$. Bounds become $1$ and $0$, so $-\int_1^0u\,du=1/2$. The negative sign and reversed bounds are two coordinated consequences of decreasing cosine; dropping both accidentally may conceal an error in a more complex problem.

Useful patterns and honest limits

The pattern $g'/g$ integrates to $\log|g|$ on nonzero intervals. The pattern $g'e^g$ integrates to $e^g$. The pattern $g'/(1+g^2)$ integrates to $\arctan g$. Recognize the derivative factor, including constant multiples, rather than merely spotting a complicated inner expression.

For $\int e^{x^2}\,dx$, choosing $u=x^2$ introduces $dx=du/(2\sqrt u)$ on a positive branch and does not produce the simple exponential integral. The missing factor $2x$ matters. A substitution that makes a formula longer is not progress, and not every elementary integrand has an elementary antiderivative. Definite integration and numerical approximation remain meaningful.

Exercises

1. Integrate $6x(3x^2+1)^4$.
2. Evaluate $\int_0^2x/(1+x^2)\,dx$.
3. Evaluate $\int_0^1 3x^2\sqrt{1+x^3}\,dx$.
4. Integrate $\cos(5x)$ and verify the factor.
5. Evaluate $\int_0^\pi\sin x\cos x\,dx$; explain why using $u=\sin x$ is still consistent despite noninjectivity.
6. Explain the missing factor in the incorrect assertion $\int e^{x^2}\,dx=e^{x^2}+C$.

Worked solutions

1. With $u=3x^2+1$, the result is $(3x^2+1)^5/5+C$.
2. $[\log(1+x^2)/2]_0^2=(\log5)/2$.
3. $u=1+x^3$ transforms the bounds to $1,2$, giving $[2u^{3/2}/3]_1^2=2(2\sqrt2-1)/3$.
4. $(1/5)\sin(5x)+C$; the chain-rule factor $5$ cancels $1/5$ on differentiation.
5. The integral is zero by $[\sin^2x/2]_0^\pi$. The substitution identity gives $\int_0^0u\,du=0$ and accounts for the orientation reversal during the path.
6. Differentiating the proposed answer gives $2xe^{x^2}$, which is a different integrand.