Mathematics
ASSERTION

$$\displaystyle \int \frac{dx}{x^{3}\sqrt{1+x^{4}}}=-\frac{1}{2}\sqrt{1+\frac{1}{x^{^{4}}}}+C.$$

REASON

For integration by parts, we have to follow ILATE rule.


ANSWER

Assertion is True, Reason is True; Reason is NOT a correct explanation for Assertion


SOLUTION
$$\displaystyle I=\int \frac{dx}{x^{3}\sqrt{1+x^{4}}}$$

$$\displaystyle =\int \frac{dx}{x^{5}\sqrt{\frac{1}{x^{^{4}}}}+1}$$

Put $$\displaystyle \frac{1}{x^{4}}+1=t$$
$$\displaystyle dt=\frac{-4}{x^{5}}dx$$

$$\displaystyle \therefore I=-\frac{1}{4}\int \frac{dt}{\sqrt{t}}$$

$$\displaystyle =-\frac{1}{2}\sqrt{t}+C$$
$$\displaystyle =-\frac{1}{2}\sqrt{1+\frac{1}{x^{4}}}+C$$
Hence, statement 1 is true.
Also, for integration by parts, we need to follow ILATE.
Thus, both the statements are true but statement 2 is not a correct explanation of statement 1.
View Full Answer

Its FREE, you're just one step away


Assertion & Reason Medium Published on 17th 09, 2020
Next Question
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84
Enroll Now For FREE

Realted Questions

Q1 Single Correct Medium
Solve :
$$\displaystyle \int x.(x^{x})^{x}(2\log x+1)dx$$ 
  • A. $$x^{(x^{x})}+c$$
  • B. $$x^{x}.\log x+c$$
  • C. does not exist
  • D. $$(x^{x})^{x}+c$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Single Correct Medium
Evaluate : $$\displaystyle\int \frac{\cos \sqrt{x}}{\sqrt{x}}dx.$$
  • A. $$\displaystyle \sin \sqrt{x}.$$
  • B. $$\displaystyle 2\sin x.$$
  • C. $$\displaystyle 2\sin \sqrt{x}/3.$$
  • D. $$\displaystyle 2\sin \sqrt{x}.$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Subjective Medium
Evaluate $$\int \frac{x- \sin x}{1- \cos x} dx$$.

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Subjective Medium
$$\int \sin x \log (\cos x)dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Passage Medium
Let $$\displaystyle f\left ( x \right )=\frac{\sin 2x \cdot \sin \left ( \dfrac{\pi }{2}\cos x \right )}{2x-\pi }$$

Then answer the following question.

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer