Definite Integration
Substitution method
Grade Class 12
Question:
<span>∫ sin<sup>-1</sup>√<span style="display:inline-block;vertical-align:middle;text-align:center;"><span style="display:block;border-bottom:1px solid black;">x</span><span style="display:block;">x+a</span></span> dx equals</span>
<span>(A) (a+x)arctan√<span style="display:inline-block;vertical-align:middle;text-align:center;"><span style="display:block;border-bottom:1px solid black;">x</span><span style="display:block;">a</span></span> - √ax + C</span>
<span>(B) (a+x)arctan√<span style="display:inline-block;vertical-align:middle;text-align:center;"><span style="display:block;border-bottom:1px solid black;">a</span><span style="display:block;">x</span></span> - √ax + C</span>
<span>(C) (a+x<sup>2</sup>)arctan√<span style="display:inline-block;vertical-align:middle;text-align:center;"><span style="display:block;border-bottom:1px solid black;">x<sup>3</sup></span><span style="display:block;">a</span></span> + √ax + C</span>
<span>(D) (a+x)arctan√<span style="display:inline-block;vertical-align:middle;text-align:center;"><span style="display:block;border-bottom:1px solid black;">x</span><span style="display:block;">a</span></span> + √ax + C</span>
Step-by-Step Solution
Key Concept: Use substitution x = a tan^2(theta) to simplify the integrand.
<span>Let x = a tan<sup>2</sup>θ, then dx = 2a tanθ sec<sup>2</sup>θ dθ. The integral becomes ∫ sin<sup>-1</sup>(tanθ) * 2a tanθ sec<sup>2</sup>θ dθ. Using integration by parts, we get (a+x)arctan√(x/a) - √ax + C.</span>
Correct Answer: 1