Definite Integration
Indefinite Integration
Grade Class 12
Question:
<span>∫ sec<sup>2</sup>θ(secθ + tanθ)<sup>2</sup>dθ</span>
<span>(secθ + tanθ)/2 [2 + tanθ(secθ + tanθ)] + C</span>
<span>(secθ + tanθ)/3 [2 + 4 tanθ(secθ + tanθ)] + C</span>
<span>(secθ + tanθ)/3 [2 + tanθ(secθ + tanθ)] + C</span>
<span>3(secθ + tanθ)/2 [2 + tanθ(secθ + tanθ)] + C</span>
Step-by-Step Solution
Key Concept: Use substitution or expand the expression and integrate term by term.
<span>Let I = ∫ sec<sup>2</sup>θ(secθ + tanθ)<sup>2</sup>dθ. Let u = secθ + tanθ. Then du = (secθtanθ + sec<sup>2</sup>θ)dθ = secθ(tanθ + secθ)dθ = secθu dθ. Thus dθ = du/(u secθ). Since secθ = (u + 1/u)/2, we can proceed with substitution or expand the integrand: sec<sup>2</sup>θ(sec<sup>2</sup>θ + tan<sup>2</sup>θ + 2secθtanθ) = sec<sup>4</sup>θ + sec<sup>2</sup>θtan<sup>2</sup>θ + 2sec<sup>3</sup>θtanθ. Integrating this leads to the result in option C.</span>
Correct Answer: C