Indefinite Integration
e^x[g(x)+g'(x)] Form — Substitution
nta_pyq_2026_jan
Grade 12
Question:
Let $f(x)=\displaystyle\int\frac{(2-x^2)\cdot e^x}{(\sqrt{1+x})(1-x)^{3/2}}\,dx$. If $f(0)=0$, then $f\!\left(\dfrac{1}{2}\right)$ is equal to:
$\sqrt{3e}-1$
$\sqrt{3e}+1$
$\sqrt{2e}+1$
$\sqrt{2e}-1$
Step-by-Step Solution
Key Concept: Write $2-x^2=(1-x)(1+x)+1$. Split integrand: $e^x\!\left(\sqrt{\tfrac{1+x}{1-x}}+\tfrac{1}{\sqrt{1+x}(1-x)^{3/2}}\right)$. Let $g(x)=\sqrt{\tfrac{1+x}{1-x}}$; then $g'(x)=\tfrac{1}{\sqrt{1+x}(1-x)^{3/2}}$. Use $\int e^x(g+g')dx=e^x g+C$.
$f(x)=e^x\sqrt{\tfrac{1+x}{1-x}}-1$. $f(1/2)=\sqrt{3e}-1$.
Correct Answer: 1