Practice Questions

Evaluate the definite integral: $\int_{0}^{1} \frac{dx}{\sqrt{4-x^2}}$

Identify the 'integrand' and the 'variable of integration' in the expression $\int (3t^2 + \cos t) dt$.

State the power rule for integration.

Evaluate the integral using a suitable substitution: $\int \frac{\cos x}{1 + \sin x} dx$

Justify why $\int_{-a}^{a} f(x) dx = 0$ if $f(x)$ is an odd function, by interpreting the definite integral as a net signed area.

Recall the standard integral of $\frac{1}{x}$.

Calculate the integral: $\int \frac{x^2+4}{x^2+1} dx$

Apply integration by parts to solve: $\int x \sec^2 x dx$

Calculate the indefinite integral: $\int (5x^4 - 2\sec^2 x + \frac{3}{x}) dx$

Define an anti derivative or primitive of a function.

A student evaluates the integral $\int_{-1}^{1} \frac{1}{x^2} dx$ as $[\frac{-1}{x}]_{-1}^{1} = (\frac{-1}{1}) - (\frac{-1}{-1}) = -1 - 1 = -2$. Critique this evaluation and justify your conclusion.

When integrating $\int x \tan^{-1}(x) dx$ by parts, justify the choice of $\tan^{-1}(x)$ as the first function.

First, prove the property $\int_{a}^{b} f(x) dx = \int_{a}^{b} f(a+b-x) dx$. Then, use this property to design a solution for evaluating $\int_{\pi/6}^{\pi/3} \frac{dx}{1+\sqrt{\cot x}}$.

Derive the formula for $\int \sqrt{a^2 - x^2} dx$ using the method of integration by parts.

Formulate the integral for finding the anti-derivative of $\sin^{-1}\left(\frac{2x}{1+x^2}\right)$ and propose a method to solve it by first simplifying the integrand.

A student is asked to evaluate $\int \frac{dx}{x(x^5+1)}$. They attempt to use partial fractions as $\frac{A}{x} + \frac{Bx^4+Cx^3+Dx^2+Ex+F}{x^5+1}$. Critique this approach. Propose a more efficient method and use it to find the correct integral.

Calculate the definite integral: $\int_{0}^{\pi/4} \tan^2 x dx$

Evaluate the integral $\int e^{2x} \cos(3x) dx$.

State the formula for integration by parts and identify which function is typically chosen as the 'first function'.

List the standard integrals for the following six special rational and irrational functions: (i) $\int \frac{dx}{x^2 - a^2}$ (ii) $\int \frac{dx}{a^2 - x^2}$ (iii) $\int \frac{dx}{x^2 + a^2}$ (iv) $\int \frac{dx}{\sqrt{x^2 - a^2}}$ (v) $\int \frac{dx}{\sqrt{a^2 - x^2}}$ (vi) $\int \frac{dx}{\sqrt{x^2 + a^2}}$

List the standard forms of partial fractions for the rational functions (i) $\frac{px^2+qx+r}{(x-a)(x-b)(x-c)}$ and (ii) $\frac{px^2+qx+r}{(x-a)^2(x-b)}$.

List the formulas for the integrals of the three special functions involving square roots.

Find the integral: $\int \frac{e^{\sqrt{x}}}{\sqrt{x}} dx$

Analyze and evaluate the integral $\int_{-\pi/2}^{\pi/2} x^3 \cos x dx$

Evaluate: $\int_{0}^{\pi/2} \frac{\sqrt{\cos x}}{\sqrt{\sin x} + \sqrt{\cos x}} dx$

Explain what the 'constant of integration' represents in an indefinite integral.

State the Second Fundamental Theorem of Integral Calculus.

Recall the standard integrals for $\int \sec x \, dx$ and $\int \operatorname{cosec} x \, dx$.

State the property of a definite integral when its upper and lower limits are the same.

Solve the integral: $\int \frac{3x+1}{(x+1)(x-2)} dx$

Calculate: $\int \frac{dx}{x^2 - 6x + 5}$

Find the integral of $e^x(\sin x + \cos x)$.

Formulate an indefinite integral of the form $\int e^x [f(x) + f'(x)] dx$ where $f(x) = \frac{1}{x^2}$, and then state its solution.

When evaluating an integral of the form $\int \frac{\cos x}{\cos x + \sin x} dx$, a common strategy is to express the numerator as $\frac{1}{2}[(\cos x + \sin x) + (\cos x - \sin x)]$. Justify why this specific algebraic manipulation is an effective strategy for solving the integral.

Demonstrate the use of integration by parts to find the integral of $\log x$.

Evaluate the integral $\int \frac{x^2+4}{x^4+16} dx$.

Calculate the definite integral: $\int_{1}^{2} \frac{5x^2}{x^2+4x+3} dx$

Derive the reduction formula for $I_n = \int \sec^n x dx$, where $n$ is a positive integer greater than 2. Use the derived formula to evaluate $\int \sec^4 x dx$.

State the property $\mathbf{P_4}$ of definite integrals and explain what it signifies.

Solve the integral: $\int \frac{x^2}{(x^2+1)(x^2+4)} dx$

Without performing explicit integration, evaluate $\int_{0}^{2\pi} \sin^7(x) dx$ and justify your reasoning using the properties of definite integrals.

Evaluate the definite integral $\int_{0}^{1} \cot^{-1}(1-x+x^2) dx$.

Explain the method of 'Integration by Substitution'. Describe the key steps involved in this process for a definite integral.

Describe the property $\mathbf{P_7}$ of definite integrals concerning even and odd functions. Explain why this property is useful.

Evaluate the definite integral $\int_{0}^{\pi} \frac{x \sin x}{1 + \sin x} dx$. Justify the key property used in your solution.