The integral \int dx \over x^2( x^4 + 1 )^3/4 equals :

- ( x^4 + 1 \over x^4 )^1 \over 4 + c

( x^4 + 1 \over x^4 )^1 \over 4 + c

Indefinite Integrals PYQ — JEE Main 2015 · Shift 0 · Q31 with Solution

View written solutionFree

Correct answer: B

We need to evaluate $I=\int \frac{dx}{x^2(x^4+1)^{3/4}}.$
We look for a substitution that simplifies the expression involving both $x^2$ and $x^4+1$ .

Notice that $\frac{x^4+1}{x^4}=1+\frac{1}{x^4}.$ So define $u=\left(\frac{x^4+1}{x^4}\right)^{1/4}=\left(1+x^{-4}\right)^{1/4}.$

First, $u=(1+x^{-4})^{1/4}.$ Therefore,

=-x^{-5}(1+x^{-4})^{-3/4}.$$ Now simplify: $$(1+x^{-4})^{-3/4}=\left(\frac{x^4+1}{x^4}\right)^{-3/4} =\frac{x^3}{(x^4+1)^{3/4}}$$ (for real $x\neq 0$, this matches the integrand form). Hence, $$\frac{du}{dx}=-x^{-5}\cdot \frac{x^3}{(x^4+1)^{3/4}} =-\frac{1}{x^2(x^4+1)^{3/4}}.$$ Thus, $$du=-\frac{dx}{x^2(x^4+1)^{3/4}}.$$ So, $$I=-\int du=-u+C.$$ 4. Substitute back: $$I=-\left(\frac{x^4+1}{x^4}\right)^{1/4}+C.$$ 5. Therefore the correct option is $$\boxed{\text{B}}.$$ 6. Quick verification by differentiation: Let $$F(x)=-\left(\frac{x^4+1}{x^4}\right)^{1/4}.$$ Then $$F'(x)=\frac{1}{x^2(x^4+1)^{3/4}},$$ which matches the integrand. Hence the answer is confirmed.

View written solutionFree

Correct answer: B

We need to evaluate $I=\int \frac{dx}{x^2(x^4+1)^{3/4}}.$
We look for a substitution that simplifies the expression involving both $x^2$ and $x^4+1$ .

Notice that $\frac{x^4+1}{x^4}=1+\frac{1}{x^4}.$ So define $u=\left(\frac{x^4+1}{x^4}\right)^{1/4}=\left(1+x^{-4}\right)^{1/4}.$

First, $u=(1+x^{-4})^{1/4}.$ Therefore,

=-x^{-5}(1+x^{-4})^{-3/4}.$$ Now simplify: $$(1+x^{-4})^{-3/4}=\left(\frac{x^4+1}{x^4}\right)^{-3/4} =\frac{x^3}{(x^4+1)^{3/4}}$$ (for real $x\neq 0$, this matches the integrand form). Hence, $$\frac{du}{dx}=-x^{-5}\cdot \frac{x^3}{(x^4+1)^{3/4}} =-\frac{1}{x^2(x^4+1)^{3/4}}.$$ Thus, $$du=-\frac{dx}{x^2(x^4+1)^{3/4}}.$$ So, $$I=-\int du=-u+C.$$ 4. Substitute back: $$I=-\left(\frac{x^4+1}{x^4}\right)^{1/4}+C.$$ 5. Therefore the correct option is $$\boxed{\text{B}}.$$ 6. Quick verification by differentiation: Let $$F(x)=-\left(\frac{x^4+1}{x^4}\right)^{1/4}.$$ Then $$F'(x)=\frac{1}{x^2(x^4+1)^{3/4}},$$ which matches the integrand. Hence the answer is confirmed.