Q.Evaluate : $\int_{0}^{\pi/4} \dfrac{\sin x + \cos x}{16 + 9\sin 2x}\, dx$ OR Evaluate : $\int_{1}^{3} (x^2 + 3x + e^x)\, dx$ as the limit of the sum.

Substitution in Definite Integrals

You already know substitution for indefinite integrals: set $u = g(x)$, rewrite in terms of $u$, integrate, then substitute back. For a definite integral there is a cleaner twist — instead of substituting back, you convert the limits of integration to the new variable and finish entirely in $u$.

Why the limits must change

The limits $a$ and $b$ are $x$-values. Once you switch to $u = g(x)$, those numbers no longer describe the start and end of the integration — the corresponding $u$-values do. Keeping the old numbers would integrate over the wrong interval, like reading a distance in kilometres off a scale marked in miles.

The steps

1. Choose $u = g(x)$, picking something whose derivative already appears in the integrand.
2. Differentiate: $du = g'(x)\,dx$.
3. Convert the limits: the lower limit becomes $u = g(a)$, the upper becomes $u = g(b)$.
4. Integrate in $u$ — no substituting back needed.

Example. Evaluate $\int_0^2 2x\,(x^2+1)^3\,dx$.

Let $u = x^2 + 1$, so $du = 2x\,dx$. When $x=0$, $u=1$; when $x=2$, $u=5$. Then

We never returned to $x$ — the converted limits carried the work. …