Home › CBSE Class 12 › Calculus › Using the substitution $t = 1 + x^2$, $\displays…
Using the substitution $t = 1 + x^2$, $\displaystyle\int \dfrac{x}{1 + x^2}\,dx$ equals:
A$\dfrac{1}{2}\log(1 + x^2) + C$
B$\dfrac{1}{2}\tan^{-1}(x^2) + C$
C$\log(1 + x^2) + C$
D$\tan^{-1} x + C$
Answer & Solution
Correct answer: A. $\dfrac{1}{2}\log(1 + x^2) + C$
With $t = 1 + x^2$, $dt = 2x\,dx$, so $\int \dfrac{x}{1+x^2}\,dx = \dfrac{1}{2}\int \dfrac{dt}{t} = \dfrac{1}{2}\log|t| + C = \dfrac{1}{2}\log(1 + x^2) + C$. Note the $1/2$ factor, which trips up answers like option B.
Related questions
A tank is draining and a student is asked how fast the depth falls when the volume is chanThe notation used for a family of antiderivatives, complete with its constant, is the:Knowing an object's velocity, a student recovers its position by finding an:A function whose derivative is the given function is called its:Calculators and computers rely on that same tangent-based idea when they find:An iterative technique for finding zeroes, built on tangent line approximations, is:The rule that resolves such a limit by differentiating numerator and denominator separatelA limit produces the form zero over zero, so its behaviour cannot be read off directly. Th