The equation of a curve passing through origin is given by y=∫x^3 cosx^4 d x. If the equation of

step 1 In this question, it is given that the equation of a curve passing through origin is given by y

The equation of a curve passing through origin is given by...

The equation of a curve passing through origin is given by $y=\int x^{3} \cos x^{4} d x$. If the equation of the curve is written in the form $x=g(y)$, then
(A) $g(y)=\sqrt[3]{\sin ^{-1}(4 y)}$
(B) $g(y)=\sqrt{\sin ^{-1}(4 y)}$
(C) $g(y)=\sqrt[4]{\sin ^{-1}(4 y)}$
(D) none of these

Key Concepts

Integration by Substitution

This is a fundamental technique in calculus used to simplify the process of evaluating integrals. It involves replacing a complicated expression with a new variable, which transforms the integral into a more manageable form. In problems involving polynomial and trigonometric functions, substitution can reduce the integrand to a standard form, facilitating its integration.

Antiderivatives and the Fundamental Theorem of Calculus

Finding the antiderivative (or indefinite integral) of a function is crucial in calculus for determining the original function whose derivative is given. The Fundamental Theorem of Calculus bridges the concept of differentiation and integration, stating that integration can be reversed by differentiation, and it allows for the incorporation of initial conditions to determine the constant of integration.

Inverse Functions

An inverse function reverses the operation of the original function, allowing one to express the input in terms of the output. In the context of curves defined by integration, once the relationship between the variables is established, the inverse function is used to express the independent variable explicitly as a function of the dependent variable.

Inverse Trigonometric Functions

These functions represent the inverse operations of trigonometric functions and are essential for solving equations where a trigonometric function of a variable is known. They enable one to isolate and solve for the variable when it appears within a trigonometric function, often after an integration or substitution process has produced such a function.

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