List of top Mathematics Questions on integral asked in MHT CET

Evaluate the indefinite integral: \[ \int e^x (\sin x + \cos x)\, dx \]

If \(\int x^3 e^x \, dx = e^x(px^3 + qx^2 + rx + s) + C\), then find the value of \(p + q + r + s\).

The value of $\int \frac{x^2-1}{(x^4+3x^2+1)\tan^{-1}(x+\frac{1}{x})} dx$ is:

The integral $\displaystyle\int \frac{1}{\sqrt[4]{(x-1)^3(x+2)^5}}\,dx$ is equal to: (where $C$ is a constant of integration)}

If $I = \int \frac{\sin x + \sin^3 x}{\cos 2x}\,dx = P \cos x + Q \log \left| \frac{\sqrt{2}\cos x - 1}{\sqrt{2}\cos x + 1} \right| + C$, then the values of $P$ and $Q$ are respectively:

The value of $\int \frac{x^2 - 1}{(x^4 + 3x^2 + 1) \tan^{-1}(x + \frac{1}{x})} dx$ is:

Let \[ f(x)=\int \frac{x\sqrt{x}}{(1+x)^2}\,dx \qquad (x\ge0) \] Then $f(3)-f(1)$ is equal to:

Evaluate the integral: \( \displaystyle \int \frac{x}{x+2}\,dx \)

Evaluate the following indefinite integral: \[ \int \sin(\log x)\,dx \]

Evaluate \( \int \log x \, dx \).

Evaluate the integral: \( \displaystyle \int \frac{4x^2 \cot^{-1}(x^3)}{1+x^6}\,dx \) (where \(C\) is a constant of integration).

A plane passes through \( (2, 1, 2) \) and \( (1, 2, 1) \) and parallel to the line \( 2x = 3y \) and \( z = 1 \), then the plane also passes through the point

$\int \frac{x dx}{(x-1)(x-2)} =$}

If $\int \tan^4 x dx = a \tan^3 x + b \tan x + cx + k$ (where k is the constant of integration) then the value of $a - b + c =$

$\int \frac{x + \sin x}{1 + \cos x} dx =$}

Which of the following are pairs of equivalent circuits

The principal increases continuously in a newly opened bank at the rate of $10%$ per year. An amount of Rs. 2000 is deposited with this bank. How much will it become after 5 years? ($e^{0.5} = 1.648$)

If \( a^2 + b^2 + c^2 = r^2 \), then the value of \( \tan^{-1}\left(\frac{ab}{cr}\right) + \tan^{-1\left(\frac{bc}{ar}\right) + \tan^{-1}\left(\frac{ca}{br}\right) = \)}

A straight line through the origin \( O \) meets the line \( 3y = 10 - 4x \) and \( 8x + 6y + 5 = 0 \) at the points \( A \) and \( B \) respectively. Then \( O \) divides the segment \( AB \) in the ratio

The value of \[ \int \frac{\sin 7x}{\cos 9x \, \cos 2x} \, dx \] is:

If \[ A= \begin{bmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{bmatrix} \] where \[ a=7^x,\quad b=7^{7^x},\quad c=7^{7^{7^x}} \] then the value of \[ \int |A| \, dx \] is:

Evaluate: \[ \int \frac{dx}{\cos x(1+\cos x)} \]

$\int \frac{dx}{2+\cos x} =$}