Exercise 1541Combinatorics › Binomial coefficientsL1Added 3 Nov 2026

Application to integration

#integration#binomial#identity

Statement

Compute $\int_0^1 (1+x)^n\,dx$ using two methods.

Hint

Direct: $\frac{2^{n+1}-1}{n+1}$ . Binomial: integrate the expansion.

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Direct. $\int_0^1(1+x)^n\,dx=\frac{(1+x)^{n+1}}{n+1}\Big|_0^1=\frac{2^{n+1}-1}{n+1}$ .

Binomial. $\int_0^1\sum_k\binom{n}{k}x^k\,dx=\sum_k\binom{n}{k}\frac{1}{k+1}$ .

Conclusion. $\sum_k\frac{\binom{n}{k}}{k+1}=\frac{2^{n+1}-1}{n+1}$ .