Evaluate the following integrals: x dx - Vidyadip

Question

Evaluate the following integrals:

$\int\cos\sqrt{\text{x}}\text{dx}$

✓

Answer

Let $\text{I}=\int\cos\sqrt{\text{x}}\text{dx}$
$\sqrt{\text{x}}=\text{t}$
$\text{x}=\text{t}^2$
$\text{dx}=2\text{t dt}$
$=\int2\text{t}\cos\text{t dt}$
$\text{I}=2\int\text{t}\cos\text{t dt}$
$\text{I}=2[\text{t}\int\cos\text{t dt}-\int(1\int\cos\text{t dt})\text{dt}]$
$=2[\text{t}\sin\text{t}-\int\sin\text{t dt}]$
$=2[\text{t}\sin\text{t}+\cos\text{t}]+\text{C}$
$\text{I}=2[\sqrt{\text{x}}\sin\sqrt{\text{x}}+\cos\sqrt{\text{x}}]+\text{C}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "2 Marks" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Find $|\vec{a} \times \vec{b}|$ if $\vec{a}=2 \hat{i}+\hat{j}+3 \hat{k}$ and $\vec{b}=3 \hat{i}+5 \hat{j}-2 \hat{k}$

For the principal values, evaluate the following:
$\cos^{-1}\frac{1}{2}+2\sin^{-1}\Big(\frac{1}{2}\Big)$

If $\vec{\text{a}}$ and $\vec{\text{b}}$ are unit vectors, then write the value of $\big|\vec{\text{a}}\times\vec{\text{b}}\big|^2+\big(\vec{\text{a}}.\vec{\text{b}}\big)^2.$

Five cards are drawn successively with replacement from a well-shuffled pack of 52 cards. What is the probability that
none is a spade?

Write the unit vector in the direction of $\vec{\text{a}}=3\hat{\text{i}}+2\hat{\text{j}}+6\hat{\text{k}}$.

If $\text{P(A)}=\frac{7}{13},\text{P(B)}=\frac{9}{13}$ and $\text{P}(\text{A}\cap\text{B})=\frac{4}{13},$ find $\text{P}\Big(\frac{\text{A}}{\text{B}}\Big).$

Write the equation of the plane parallel to XOY- plane and passing through the point (2, -3, 5).

Six coins are tossed simultaneously. Find the probability of getting.
no heads.

Find the value of k so that the lines x = -y = kz and x - 2 = 2y + 1 = -z + 1 are perpendicular to each other.

If R is a symmetric relation on a set A, then write a relation between R and R^-1.