Integral Calculus Spring 2024 Sheet (6)

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What is the net area between the curve of $f(x)$ and the $x$-axis in the interval $[-3, 7]$?

The net area is the total area minus the area under the $x$-axis, which needs to be calculated by integrating the function over the given interval.

Evaluate the integral: $\int \sin^4 x \cos^2 x , dx$

Using the trigonometric substitution $u = \sin x$, the integral becomes $\int u^4 (1 - u^2) , du$, which can be solved using the technique of integration by parts.

Evaluate the integral: $\int \frac{x \tan^{-1} x}{\sqrt{1 + x^2}} , dx$

This integral can be solved using the trigonometric substitution $x = \tan \theta$, which leads to an integral involving powers of sine and cosine functions.

Evaluate the integral: $\int \frac{\sec^2 \sqrt{x}}{\sqrt{x}} , dx$

This integral falls under the category of the second generalization rule, where the substitution $u = \sqrt{x}$ can be used to simplify the integral. Signup and view all the answers

Use integration by parts to evaluate: $\int x \cos x , dx$

Let $u = x$ and $dv = \cos x , dx$. Then $du = dx$ and $v = \sin x$. Applying the integration by parts formula yields the solution. Signup and view all the answers

Find the value of the integral: $\int_0^{\pi/2} \cos^2 x , dx$

Using the trigonometric identity $\cos^2 x = \frac{1}{2} + \frac{1}{2} \cos 2x$, and evaluating the integral over the given limits yields the solution. Signup and view all the answers

Evaluate the integral: $\int \frac{dx}{x^4 \sqrt{x^2 + 3}}$

This integral can be solved using the trigonometric substitution $x = \sqrt{3} \tan \theta$, which transforms the integral into one involving powers of sine and cosine functions. Signup and view all the answers

Use integration by parts to evaluate: $\int x \ln x , dx$

Let $u = \ln x$ and $dv = x , dx$. Then $du = \frac{1}{x} , dx$ and $v = \frac{1}{2} x^2$. Applying the integration by parts formula yields the solution. Signup and view all the answers

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

This integral can be solved using the trigonometric substitution $x = \tan \theta$, which transforms the integral into one involving powers of sine and cosine functions. Signup and view all the answers

Find the value of the integral: $\int \sqrt{x^2 - 2} , dx$

This integral can be solved using the trigonometric substitution $x = \sqrt{2} \sec \theta$, which transforms the integral into one involving powers of secant and tangent functions. Signup and view all the answers